# On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces

Authors: A. Lastra; S. Malek
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with 2π-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis-Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.


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## Body

## 1. Introduction

We consider a family of nonlinear Cauchy problems of the form(1)ϵr3t2∂t+tr2+-i∂z+1r1∂xSXit,z,x,ϵ=∑k_=(s,k0,k1,k2)∈Sbk_z,x,ϵts∂tk0∂zk1∂xk2Xit,z,x,ϵ+∑l_=(l0,l1)∈Ncl_z,x,ϵtl0+l1-1Xit,z,x,ϵl1
for given initial data
(2)∂xjXit,z,0,ϵ=Ξi,jt,z,ϵ,ii0≤i≤ν-1,0≤j≤S-1,
where ϵ is a complex parameter, S, r1, r2, r3 are some positive integers, S is a finite subset of N4, and N is a finite subset of N2 that fulfills constraints (185). The coefficients bk_(z,x,ϵ) of the linear part and cl_(z,x,ϵ) of the nonlinear part are 2π-periodic Fourier series
(3)bk_z,x,ϵ=∑β≥0bk_,βx,ϵeizβ,cl_z,x,ϵ=∑β≥0cl_,βx,ϵeizβ
in the variable z with coefficients bk_,β(x,ϵ), cl_,β(x,ϵ) in O{x,ϵ} (which denotes the Banach space of bounded holomorphic functions in (x,ϵ) on some small polydisc D(0,ρ)×D(0,ϵ0) centered at the origin in C2 with supremum norm). We assume that all Fourier coefficients bk_,β(x,ϵ), cl_,β(x,ϵ) have exponential decay in β (see (177), (178)). Hence, bk_(z,x,ϵ) and cl_(z,x,ϵ) define bounded holomorphic functions on Hρ′×D(0,ρ)×D(0,ϵ0) where Hρ′={z∈C/Im⁡(z)<ρ′} is some strip of width 2ρ′>0.The initial data are quasiperiodic in the variablez and are constructed as follows:
(4)Ξi,jt,z,ϵ=∑β0,…,βl≥0Ξi,β0,…,βl,jt,ϵexp⁡⁡iz∑j=0lβjξjβ0!⋯βl!,
where ξ0=1 and ξ1,…,ξl are real algebraic numbers (for some integer l≥1) such that the family {1,ξ1,…,ξl} is Z-linearly independent and where the coefficients Ξi,β0,…,βl,j(t,ϵ) are bounded holomorphic functions on T×Ei, where T is a fixed open bounded sector centered at 0 and E_={Ei}0≤i≤ν-1 is a family of open bounded sectors centered at the origin and whose union forms a covering of V∖{0}, where V denotes some bounded neighborhood of 0. These functions (4) are constructed in such a way that they define holomorphic functions on T×Hρ1′×Ei for some 0<ρ1′<ρ′.Recall that a functionf:R→E (where E denotes some vector space) is said to bequasiperiodic with period w=(w0,…,wl)∈Rl+1, for some integer l≥1, if there exists a function F:Rl+1→E such that, for all 0≤j≤l, the partial function xj↦F(x0,…,xj,…,xl) is wj-periodic on R, for all fixed xk∈R when k≠j, which satisfies f(t)=F(t,…,t) (see, e.g., [1] for a definition and properties of quasiperiodic functions). In particular, one can check that functions (4) are quasiperiodic with period w=(2π,2π/ξ1,…,2π/ξl) in the z variable.Our main purpose is the construction of actual holomorphic functionsXi(t,z,x,ϵ) to the problem (1), (2) on the domains T×Hρ1′×D(0,ρ1)×Ei for some small disc D(0,ρ1)⊂C and the analysis of their asymptotic expansions as ϵ tends to zero on Ei, for all 0≤i≤ν-1. More precisely, we can state our main result as follows.Main Statement. We take a set of directions
d
i
∈
R
,
0
≤
i
≤
ν
-
1
, such that
d
i
≠
π
2
k
+
1
/
r
2
, for
0
≤
k
≤
r
2
-
1
, which are assumed to satisfy moreover
(5)
r
3
r
2
arg
ϵ
+
arg
t
∈
d
i
-
θ
,
d
i
+
θ
for all
ϵ
∈
E
i
, all
t
∈
T
, and all
0
≤
i
≤
ν
-
1
, for some fixed
θ
>
π
. We make the hypothesis that the coefficients
Ξ
i
,
β
0
,
…
,
β
l
,
j
(
t
,
ϵ
)
of the initial data (4) can be expressed as Laplace transforms
(6)
Ξ
i
,
β
0
,
…
,
β
l
,
j
t
,
ϵ
=
1
ϵ
r
3
/
r
2
t
∫
L
d
i
V
i
,
β
0
,
…
,
β
l
,
j
τ
,
ϵ
e
-
τ
/
ϵ
r
3
/
r
2
t
d
τ
on
T
×
E
i
along the half-line
L
d
i
=
R
+
e
-
1
d
i
, where
V
i
,
β
0
,
…
,
β
l
,
j
(
τ
,
ϵ
)
is a family of holomorphic functions which share the exponential growth constraints (152) with respect to
τ
, the uniform bound estimates (161), and the analytic continuation property (184).Then, in Proposition28, we construct a family of holomorphic and bounded functions
(7)
X
i
t
,
z
,
x
,
ϵ
=
∑
β
0
,
…
,
β
l
≥
0
X
i
,
β
0
,
…
,
β
l
t
,
x
,
ϵ
exp
⁡
⁡
i
z
∑
j
=
0
l
β
j
ξ
j
β
0
!
⋯
β
l
!
which are quasiperiodic with period
w
=
(
2
π
,
2
π
/
ξ
1
,
…
,
2
π
/
ξ
l
)
in the variable
z
and which solve the problem (1), (2) on the products
T
×
H
ρ
1
′
×
D
(
0
,
ρ
1
)
×
E
i
, where
ρ
1
′
>
0
satisfies inequality (153) and for some small radius
0
<
ρ
1
<
ρ
. Moreover, the differences
X
i
+
1
(
t
,
z
,
x
,
ϵ
)
-
X
i
(
t
,
z
,
x
,
ϵ
)
satisfy the exponential decay (187) whose type depends on the constants
r
1
,
r
2
,
r
3
and on the degree
h
+
1
of any algebraic number field
Q
(
ξ
)
containing
ξ
1
,
…
,
ξ
l
.In Theorem32, we show the existence of a formal series
(8)
X
^
ϵ
=
∑
k
≥
0
H
k
t
,
z
,
x
ϵ
k
k
!
whose coefficients
H
k
(
t
,
z
,
x
)
belong to the Banach space of bounded holomorphic functions on
T
×
H
ρ
1
′
×
D
(
0
,
ρ
1
)
, which formally solves (1) and is moreover the Gevrey asymptotic expansion of order
h
r
1
+
r
2
/
r
3
of
X
i
on
E
i
. In other words, there exist two constants
C
,
M
>
0
such that
(9)
sup
⁡
t
∈
T
,
z
∈
H
ρ
1
′
,
x
∈
D
(
0
,
ρ
1
)
X
i
t
,
z
,
x
,
ϵ
-
∑
k
=
0
N
-
1
H
k
t
,
z
,
x
ϵ
k
k
!
≤
C
M
N
N
!
(
h
r
1
+
r
2
)
/
r
3
ϵ
N
for all
N
≥
1
and all
ϵ
∈
E
i
.Notice that the problem (1), (2) is singularly perturbed with irregular singularity (in the sense of Mandai, [2]) with respect to t at t=0 provided that r2>r1. It is of Kowalevski type if r2<r1 (meaning that the hypotheses of the classical Cauchy-Kowalevski theorem (see, e.g., [3], pp. 346–349) are fulfilled for (1)) and of mixed type when r2 and r1 are equal.In a recent work [4], we have considered singularly perturbed nonlinear Cauchy problems of the form
(10)ϵr3z∂zr1t2∂tr2∂zSuit,z,ϵ=Ft,z,ϵ,∂t,∂zuit,z,ϵ+Pt,z,ϵ,uit,z,ϵ
which carry both an irregular singularity with respect to t at t=0 and a Fuchsian singularity (see [5] for a definition) with respect to z at z=0, for given initial data
(11)∂zjuit,0,ϵ=ϕi,jt,ϵ,ii0≤i≤ν-1,0≤j≤S-1,
where F is some linear differential operator with polynomial coefficients and P is some polynomial. The initial data ϕj,i(t,ϵ) were assumed to be holomorphic on products T×Ei. Under suitable constraints on the shape of (10) and on the initial data (11), we have shown the existence of a formal series u^(ϵ)=∑k≥0‍hkϵk/k! with coefficients hk belonging to the Banach space F of bounded holomorphic functions on T×D(0,δ) (for some δ>0) equipped with the supremum norm, solution of (10), which is the Gevrey asymptotic expansion of order r1+r2/r3 of actual holomorphic solutions ui of (10), (11) on Ei as F-valued functions, for all 0≤i≤ν-1.Compared to this former result [4], the singularity nature of (1) does not come from the divergence of the formal series. This divergence rather emerges from the quasiperiodic structure of the solution space which produces a small divisor problem (as we will see below) and its Gevrey type depends not only on the type of space of our initial data but also on the shape of (1). It is worth noticing that a similar phenomenon has been observed in the paper [6] for the steady Swift-Hohenberg equation
(12)1+Δ2Ux,μ-μUx,μ+U3x,μ=0,
where the authors have constructed formal series solutions
(13)Ux,μ=μ∑n≥0Unxμn,
where the coefficients U(n)(x) belong to some weighted Sobolev space Hs(Γ) (for well-chosen real number s>0) of quasiperiodic Fourier expansions in x∈R2 of the form
(14)Unx=∑k∈ΓUkeik·x,
where Γ={∑j=1Q‍mjkj/(m1,…,mQ)∈NQ} with kj=(cos⁡(2π((j-1)/Q)),sin(2π((j-1)/Q))) is the so-called quasilattice in R2 for some integer Q≥8. They have shown that this formal series (13) is actually at most of Gevrey order 4l (for a suitable integer l depending on Q) as series in the Hilbert space Hs(Γ). Their main purpose was actually to use this result in order to construct approximate smooth quasiperiodic solutions of (12) up to an exponential small order by means of truncated Laplace transforms.In a more general setting, the Cauchy problem (1), (2) we consider in this work comes within the framework of asymptotic analysis of solutions to differential equations or to partial differential equations with periodic or quasiperiodic coefficients which is a domain of intense research in these last years.In the category of differential equations most of the results concern nonlinear equations of the form(15)∑k=0Kakϵ∂tkut,ϵ=Fut,ϵ,t,ϵ,
where the forcing term F contains periodic or quasiperiodic coefficients. These statements deal with the construction of formal solutions u(t,ϵ)=∑l=0+∞‍ul(t)ϵl which are called Lindstedt series in the literature. For convergence properties of these series, we quote the seminal work [7] and the overview [8], and for Borel resummation procedures applied more recently, we mention [9]. For applications in KAM theory for nearly integrable finitely dimensional Hamiltonian systems, we may refer to [10, 11].In the context of partial differential equations, for existence results of quasiperiodic solutions to general families of nonlinear PDE containing a small real parameter, we indicate [12] and for the construction of periodic solutions to abstract second order nonlinear equations, we notice [13]. Concerning KAM theory results in the context of PDE such as small nonlinear perturbations of wave equations or Schrödinger equations we mention the fundamental works [14–16].Now, we explain our main result and the principal arguments needed in its proof. The first step consists (as in [4]) of transforming (1) by means of the linear map T↦T/ϵr3/r2 into an auxiliary regularly perturbed nonlinear equation (149). The drawback of this transformation is the appearance of poles in the coefficients of this new equation with respect to ϵ at 0.The approach we follow is the same as in our previous works [4, 17] and is based on a Borel resummation procedure applied to formal expansions of the form
(16)Y^T,z,x,ϵ=∑β_=(β0,…,βl,βl+1)∈Nl+2Y^β_T,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!,
where Y^β_(T,ϵ)=∑m≥0‍χm,β_(ϵ)Tm/m! are formal series in T, which formally solves the auxiliary equation (149) for well-chosen initial data (165). It is worth pointing out that this resummation method known as κ-summability already enjoys a large success in the study of Gevrey asymptotics for analytic solutions to linear and nonlinear differential equations with irregular singularity; see, for instance, [18–24]. We show that the formal Borel transform of Y^(T,z,x,ϵ) with respect to T given by
(17)V^τ,z,x,ϵ=∑β_=(β0,…,βl,βl+1)∈Nl+2V^β_τ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!,
where V^β_(τ,ϵ)=∑m≥0‍χm,β_(ϵ)τm/(m!)2, formally solves a nonlinear convolution integrodifferential Cauchy problem with rational coefficients in τ and is holomorphic with respect to x near the origin and with respect to z in some strip and meromorphic in ϵ with a pole at 0; see (171), (172).For appropriate initial data satisfying conditions (152), (184), and (161), we show (in Proposition 20) that the formal series V^(τ,z,x,ϵ) actually defines a holomorphic function Vi on the product Ui×Hρ1′×D(0,ρ1)×D(0,ϵ0)∖{0}, for some 0<ρ1′<ρ′, 0<ρ1<ρ and where Ui is some unbounded open sector with small aperture and with bisecting direction di (as described above in the main statement). The functions Vi have exponential growth rate with respect to (τ,ϵ) meaning that there exist two constants C,K>0 such that
(18)sup⁡z∈Hρ1′,x∈D(0,ρ1)Viτ,z,x,ϵ≤CeKτ/ϵ
for all τ∈Ui, ϵ∈D(0,ϵ0)∖{0}. Moreover, we show that, for all β_=(β0,…,βl,βl+1)∈Nl+2, the formal series V^β_(τ,ϵ) actually define holomorphic functions Vi,β_(τ,ϵ) on domains (Ui∪D(0,ρβ_))×D(0,ϵ0)∖{0}, where ρβ_ is a Riemann type sequence of the form R/(1+|β_|)hr1/r2, for some constant R>0, which tends to 0 as |β_|=∑j=0l+1‍βj tends to infinity, and share the same exponential growth rate, namely, that there exist constants C>0, K>0, M>0 with
(19)sup⁡z∈Hρ1′,x∈D(0,ρ1)Vi,β_τ,ϵ≤CK∑j=0l+1βjβ0!⋯βl+1!eMτ/ϵ
for all τ∈Ui, ϵ∈D(0,ϵ0)∖{0}. We point out that the occurrence of a radius of convergence shrinking to zero for the coefficients Vi,β_ near the origin of the Borel transform is due to the presence of a small divisor phenomenon in the convolution Cauchy problem (171), (172) mentioned above. In our previous study [4], a similar outcome was caused by a leading term in the main equation (10) containing a Fuchsian operator (z∂z)r1. In this analysis, the denominators arise from the function space where the solutions are found, especially from their Fourier exponents ∑j=0l+1‍βjξj which may tend to zero but not faster than a Riemann type sequence as follows from Lemma 13.In order to get the estimates described above, we use a majorizing technique described in Propositions17, 18, and 19 which reduces the investigation for bounds (19) to the study of a Cauchy-Kowalevski type problem (114), (115) in several complex variables for which local analytic solutions are found in Section 2.1; see Proposition 5. On the way we make use of estimates in weighted Banach spaces introduced in Section 2.2; see Propositions 9, 10, and 12 and Corollary 11, which are very much like those already seen in the work [4].In the next step, for given suitable initial data (150) satisfying (158), we construct actual solutions
(20)YiT,z,x,ϵ=∑β_=(β0,…,βl,βl+1)∈Nl+2Yi,β_T,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!
of (149), where each function Yi,β_(T,ϵ) can be written as a Laplace transform of the function Vi,β_(τ,ϵ) with respect to τ along a half-line Lγi=R+ei-1γi⊂Ui∪{0}. For each ϵ∈Ei, the function T↦Yi,β_(T,ϵ) is bounded and holomorphic on a sector Ui,ϵ with aperture larger than π, with bisecting direction γi and with radius h′|ϵ|r3/r2 for some constant h′>0. In Proposition 23, we show that the function Yi itself turns out to define a holomorphic function on Ui,ϵ×Hρ1′×D(0,ρ1) for some 0<ρ1<ρ and where ρ1′>0 satisfies (153).We observe that, for all0≤i≤ν-1, the functions Xi defined as
(21)Xit,z,x,ϵ=Yiϵr3/r2t,z,x,ϵ
actually solve our initial Cauchy problem (1), (2) on the products T×Hρ1′×D(0,ρ1)×Ei and bear representation (7) as a quasiperiodic function whose Fourier coefficients decay exponentially in β_. It is worthy to mention that spaces of quasiperiodic Fourier series with exponential decay were also recently used in [25] in order to find global in time and quasiperiodic in space solutions to the KdV equation.In Proposition28, we show moreover that the difference of two neighboring solutions Xi+1 and Xi has exponentially small bounds of order r3/hr1+r2, uniformly in (t,z,x), as ϵ tends to 0 on Ei+1∩Ei. We observe that for each β_∈Nl+2 the difference Xi+1,β_-Xi,β_ for the Fourier coefficients has exponential decay of order r3/r2 but its type is proportional to ρβ_ and therefore tends to 0 as β_ tends to infinity. This small denominator phenomenon is the reason for the decreasement of the order r3/r2 to r3/hr1+r2. As in our previous study [4], the bulk of the proof rests on a thorough estimation of a Dirichlet like series of the form ∑k≥0‍e-1/(k+1)αϵrak for 0<a<1 and α,r>0 with ϵ>0 small. This kind of series appears in the context of almost periodic functions introduced by H. Bohr; see, for instance, the textbook [26]. These estimates (187) are crucial in order to apply a cohomological criterion known in the literature as the Ramis-Sibuya theorem (Theorem RS) which leads to the main result of this paper, namely, the existence of a formal series
(22)X^t,z,x,ϵ=∑k≥0Hkϵkk!
with coefficients Hk in the Banach space of holomorphic and bounded functions on T×Hρ1′×D(0,ρ1), which formally solves (1) and which is, moreover, the Gevrey asymptotic expansion of order hr1+r2/r3 of the functions Xi on Ei, for all 0≤i≤ν-1.The layout of the paper reads as follows.Section2.1 is dedicated to the study of a version of the Cauchy-Kowalevski theorem for nonlinear PDEs in analytic spaces of functions with precise control on the domain of existence of their solutions in terms of norm estimates of the initial data. In Section 2.2 we establish some continuity properties of several integrodifferential and multiplication operators acting on weighted Banach spaces of holomorphic functions. These results are applied in Section 2.3 when looking for global solutions with growth constraint at infinity for a parameter depending nonlinear convolution differential Cauchy problem with singular coefficients.We recall briefly the classical theory concerning the Borel-Laplace transform and we show some commutation formulas with multiplication and integrodifferential operators in Section3.1; then we center our attention on finding solutions of an auxiliary nonlinear Cauchy problem obtained by the linear change of variable T↦T/ϵr from our main Cauchy problem in Section 3.2. The link between this Cauchy problem and the one solved in Section 2.3 is performed by means of Borel-Laplace transforms on the corresponding solutions.In Section4.1, we construct actual holomorphic solutions Xi, 0≤i≤ν-1, of our initial problem and we show exponential decay of the difference of any two of these solutions with respect to ϵ on the intersection of their domain of definition, uniformly in the other variables. Finally, in Section 4.2, we conclude with the main result of the work, that is, the existence of a formal power series with coefficients in an appropriate Banach space, which asymptotically represents the functions Xi with a precise control on the Gevrey order on the sectors Ei, for all 0≤i≤ν-1.


## 2. A Global Cauchy Problem in Holomorphic and Quasiperiodic Function Spaces

### 2.1. A Cauchy-Kowalevski Theorem in Several Variables

In this section, we recall the well-known Cauchy-Kowalevski theorem in some spaces of analytic functions for which the size of the domain of existence of the solution can be controlled in terms of some supremum norm of the initial data.The next Banach spaces are natural extensions to the several variables cases of the spaces used in [27].Definition 1.
Letl≥1 be some integer. Let Z¯0,…,Z¯l,X¯>0 be positive real numbers. We denote by G(Z¯0,…,Z¯l,X¯) the space of formal series
(23)UZ0,…,Zl,X=∑β_=(β0,…,βl,βl+1)∈Nl+2uβ_Z0β0⋯ZlβlXβl+1β0!⋯βl!βl+1!
that belong to C[[Z0,…,Zl,X]] such that
(24)UZ0,…,Zl,XZ¯0,…,Z¯l,X¯=∑β_=(β0,…,βl,βl+1)∈Nl+2uβ_Z¯0β0⋯Z¯lβlX¯βl+1∑j=0l+1βj!
is finite. One can show that G(Z¯0,…,Z¯l,X¯) equipped with the norm ·(Z¯0,…,Z¯l,X¯) are Banach spaces.In the next two lemmas we show continuity properties for some linear integrodifferential operators acting on the aforementioned Banach spaces.Lemma 2.
Leth0,…,hl,hl+1∈N with
(25)hl+1≥∑j=0lhj.
Then, for any given Z¯0,…,Z¯l,X¯>0, the operator ∂Z0h0⋯∂Zlhl∂X-hl+1 is a bounded linear map from G(Z¯0,…,Z¯l,X¯) into itself. Moreover,
(26)∂Z0h0⋯∂Zlhl∂X-hl+1UZ0,…,Zl,X(Z¯0,…,Z¯l,X¯)≤Z¯0-h0⋯Z¯l-hlX¯hl+1UZ0,…,Zl,X(Z¯0,…,Z¯l,X¯)
for all U(Z0,…,Zl,X)∈G(Z¯0,…,Z¯l,X¯).Proof.
LetU(Z0,…,Zl,X)∈G(Z¯0,…,Z¯l,X¯) of form (23). By definition, we can write
(27)∂Z0h0⋯∂Zlhl∂X-hl+1UZ0,…,Zl,X(Z¯0,…,Z¯l,X¯)=∑β0,…,βl,βl+1≥0∑j=0lβj+hj+βl+1-hl+1!∑j=0l+1βj!mmmmmmmmmi·∑j=0lβj+hj+βl+1-hl+1!∑j=0l+1βj!Z¯0-h0⋯Z¯l-hlX¯hl+1·Uβ0+h0,…,βl+hl,βl+1-hl+1∑j=0lβj+hj+βl+1-hl+1!·Z¯0β0+h0⋯Z¯lβl+hlX¯βl+1-hl+1.From (25), we have
(28)∑j=0lβj+hj+βl+1-hl+1!∑j=0l+1βj!≤1
for all β0,h0,…,βl,hl,βl+1,hl+1≥0. Estimate (26) follows.Lemma 3.
Leth0,…,hl,hl+1∈N. Let, for all 0≤j≤l, 0<Z¯j1<Z¯j0 and 0<X¯1<X¯0 be positive real numbers. Then, there exists C2>0 (depending on h0,…,hl+1, X¯0,X¯1, Z¯j0,Z¯j1 for 0≤j≤l) such that
(29)∂Z0h0⋯∂Zlhl∂Xhl+1UZ0,…,Zl,X(Z¯01,…,Z¯l1,X¯1)≤C2Z¯01-h0⋯Z¯l1-hlX¯1-hl+1·UZ0,…,Zl,X(Z¯00,…,Z¯l0,X¯0)
for all U(Z0,…,Zl,X)∈G(Z¯00,…,Z¯l0,X¯0).Proof.
LetU(Z0,…,Zl,X)∈G(Z¯00,…,Z¯l0,X¯0) be of form (23). By definition, we can write
(30)∂Z0h0⋯∂Zlhl∂Xhl+1UZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1=∑β0,…,βl,βl+1≥0uβ0+h0,…,βl+hl,βl+1+hl+1∑j=0l+1βj+hj!hhhhhhhhh·Z¯01-h0⋯Z¯l1-hlX¯1-hl+1hhhhhhhhh·∑j=0l+1βj+hj!∑j=0l+1βj!×Z¯01β0+h0⋯Z¯l1βl+hlhhhhhhhhhhh·X¯1βl+1+hl+1∑j=0l+1βj+hj!∑j=0l+1βj!.
Now, we take some real number 0<δ<1 such that Z¯j1<δZ¯j0 for all 0≤j≤l and X¯1<δX¯0. We get the estimates
(31)∑j=0l+1βj+hj!∑j=0l+1βj!Z¯01β0+h0⋯Z¯l1βl+hlX¯1βl+1+hl+1≤∑j=0l+1βj+∑j=0l+1hj(∑j=0l+1hj)δ(∑j=0l+1βj+∑j=0l+1hj)i·Z¯00β0+h0⋯Z¯l0βl+hlX¯0βl+1+hl+1
for all β0,h0,…,βl,hl,βl+1,hl+1≥0. Since 0<δ<1, we know that, for any real number a>0, there exists Kδ,a>0 depending on a,δ such that
(32)sup⁡n≥0n+aaδn+a≤Kδ,a.
Hence, we get a constant C2>0 depending on hj, 0≤j≤l+1 and δ with
(33)∑j=0l+1βj+∑j=0l+1hj(∑j=0l+1hj)δ(∑j=0l+1βj+∑j=0l+1hj)·Z¯00β0+h0⋯Z¯l0βl+hlX¯0βl+1+hl+1≤C2Z¯00β0+h0⋯Z¯l0βl+hlX¯0βl+1+hl+1
for all β0,h0,…,βl,hl,βl+1,hl+1≥0. As a result, gathering (30), (31), and (33), we get the lemma.Lemma 4.
LetU1(Z0,…,Zl,X),U2(Z0,…,Zl,X)∈G(Z¯0,…,Z¯l,X¯). Then, the product U1U2 belongs to G(Z¯0,…,Z¯l,X¯). Moreover, we have
(34)U1Z0,…,Zl,XU2Z0,…,Zl,X(Z¯0,…,Z¯l,X¯)≤U1Z0,…,Zl,X(Z¯0,…,Z¯l,X¯)·U2Z0,…,Zl,X(Z¯0,…,Z¯l,X¯).
In other words, the space G(Z¯0,…,Z¯l,X¯) is a Banach algebra.Proof.
Let(35)UjZ0,…,Zl,X=∑β0,…,βl,βl+1≥0uβjZ0β0⋯ZlβlXβl+1β0!⋯βl!βl+1!
belonging to G(Z¯0,…,Z¯l,X¯) for j=1,2. By definition, we can write
(36)U1Z0,…,Zl,XU2Z0,…,Zl,XZ¯0,…,Z¯l,X¯≤∑β0,…,βl,βl+1≥0∑βj1+βj2=βj,0≤j≤l+1Πj=0l+1βj!Πj=0l+1βj1Πj=0l+1βj2!1∑j=0l+1βj!mmmmmmmmm·∑βj1+βj2=βj,0≤j≤l+1Πj=0l+1βj!Πj=0l+1βj1Πj=0l+1βj2!1∑j=0l+1βj!uβ01,…,βl1,βl+111uβ02,…,βl2,βl+122·Z¯0β0⋯Z¯lβlX¯βl+1.
Besides, using the identity (1+x)∑j=0l+1‍βj=Πj=0l+1(1+x)βj and the binomial formula, we get that
(37)Πj=0l+1βj!Πj=0l+1βj1!Πj=0l+1βj2!≤∑j=0l+1βj!∑j=0l+1βj1!∑j=0l+1βj2!
for all integers βj,βj1,βj2≥0 such that βj=βj1+βj2, for 0≤j≤l+1. Therefore, inequality (34) follows from (36) and (37).In the next proposition, we state a version of the Cauchy-Kowalevski theorem.Proposition 5.
LetD1(resp., D2) be a finite subset of Nl+2(resp., Nl+1) and let S≥1 be an integer such that, for all (k0,…,kl+1)∈D1 and all (p0,…,pl)∈D2, we have
(38)S>kl+1,S≥∑j=0l+1kj,S≥∑j=0lpj.
Let D3 be a finite subset of N∖{0,1}. Let M0,X¯0>0 be given real numbers and let Z¯j0>M0, 0≤j≤l be real numbers. Let
(39)dk0,…,kl+1Z0,…,Zl,X∈GZ¯00,…,Z¯l0,X¯0,fp0,…,plZ0,…,Zl∈GZ¯00,…,Z¯l0,X¯0,emZ0,…,Zl,X∈GZ¯00,…,Z¯l0,X¯0
for all (k0,…,kl+1)∈D1, all (p0,…,pl)∈D2, and all m∈D3. For all 0≤j≤S-1, we also choose φj(Z0,…,Zl)∈G(Z¯00,…,Z¯l0,X¯0).
We consider the following Cauchy problem:(40)∂XSUZ0,…,Zl,X=∑k_=(k0,…,kl,kl+1)∈D1dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1·UZ0,…,Zl,X+∑p_=(p0,…,pl)∈D2fp_Z0,…,Zl∂Z0p0⋯∂Zlpl·∑m∈D3emZ0,…,Zl,XUZ0,…,Zl,Xm
for given initial data
(41)∂XjUZ0,…,Zl,0=φjZ0,…,Zl,0≤j≤S-1.
Then, for given real numbers Z¯j1>0, 0≤j≤l, with M0<Z¯j1<Z¯j0, one can choose 0<X¯1<X¯0 and δ>0  (which depend on Z¯j1 for 0≤j≤l, on dk_Z0,…,Zl,X(Z¯00,…,Z¯l0,X¯0) for k_∈D1, on fp_(Z0,…,Zl)(Z¯00,…,Z¯l0,X¯0) for p_∈D2, and on em(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for m∈D3) such that if
(42)φjZ0,…,ZlZ¯00,…,Z¯l0,X¯0<δ,0≤j≤S-1,
then the problem (40), (41) has a unique solution U(Z0,…,Zl,X)∈G(Z¯01,…,Z¯l1,X¯1). Moreover, there exists a constant C3>0, depending on X¯0,X¯1,Z¯j0,Z¯j1 for 0≤j≤l, on dk_(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for k_∈D1, on fp_(Z0,…,Zl)(Z¯00,…,Z¯l0,X¯0) for p_∈D2, and on em(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for m∈D3, such that
(43)UZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤δC3.Proof. We put
(44)wZ0,…,Zl,X=∑j=0S-1φjZ0,…,ZlXjj!
and we consider the map A:C[[Z0,…,Zl,X]]→C[[Z0,…,Zl,X]] defined as
(45)AHZ0,…,Zl,X=∑k_=(k0,…,kl,kl+1)∈D1dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1-S·HZ0,…,Zl,X+∑k_=(k0,…,kl,kl+1)∈D1dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1·wZ0,…,Zl,X+∑p_=(p0,…,pl)∈D2fp_Z0,…,Zl∂Z0p0⋯∂Zlpl·∑m∈D3emZ0,…,Zl,Xmii·∑m∈D3emZ0,…,Zl,X∂X-SHZ0,…,Zl,X+wZ0,…,Zl,Xm.Lemma 6.
LetZ¯j1>0, 0≤j≤l, be real numbers such that M0<Z¯j1<Zj0. Then, there exist δ>0, a real number 0<X¯1<X¯0, and a constant K4>0  (depending on Z¯j1 for 0≤j≤l, on dk_(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for k_∈D1, on fp_(Z0,…,Zl)(Z¯00,…,Z¯l0,X¯0) for p_∈D2, and on em(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for m∈D3), such that if one puts R=K4δ,(i)
we have(46)ABR⊂BR,
where BR is the closed ball of radius R, centered at 0 in G(Z¯01,…,Z¯l1,X¯1);(ii)
for allH1,H2∈BR,
(47)AH1-AH2Z¯01,…,Z¯l1,X¯1≤12H1-H2Z¯01,…,Z¯l1,X¯1.Proof.
We first show (i). We fixZ¯j1>0, 0≤j≤l, be real numbers such that M0<Z¯j1<Zj0. We also consider δ>0 for which (42) holds. Let R>0 be of the form R=K4δ for some constant K4>0. We take H∈BR where BR is the closed ball of radius R, centered at 0 in G(Z¯01,…,Z¯l1,X¯1) for some real number 0<X¯1<X¯0. From Lemmas 2 and 4, we get
(48)dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1-Sm·∂Xkl+1-SHZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1·Z¯01-k0⋯Z¯l1-klX¯1S-kl+1·HZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯00,…,Z¯l0,X¯0·Z¯01-k0⋯Z¯l1-kl×X¯1S-kl+1R.
Now, from Lemmas 3 and 4, we get a constant C4>0 (depending on k0,…,kl, X¯0,X¯1,Z¯j0,Z¯j1, for 0≤j≤l) with
(49)dk_Z0,…,Zl,X∂Z0k0⋯∂ZlklφjZ0,…,ZlXqq!Z¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1Xqq!Z¯01,…,Z¯l1,X¯1·C4Z¯01-k0⋯Z¯l1-kl×φjZ0,…,ZlZ¯00,…,Z¯l0,X¯0≤dk_Z0,…,Zl,XZ¯00,…,Z¯l0,X¯0Xqq!Z¯00,…,Z¯l0,X¯0·C4Z¯01-k0⋯Z¯l1-klδ
for all 0≤j, q≤S-1.
Using Leibniz formula, we deduce from Lemma4 that
(50)fp_Z0,…,Zp∂Z0p0⋯∂Zlplj·emZ0,…,Zl,X∂X-SHZ0,…,Zl,X+wZ0,…,Zl,Xm1m·∂X-SHZ0,…,Zl,X+wZ0,…,Zl,XmZ¯01,…,Z¯l1,X¯1≤∑∑j=0mph,j=ph,0≤h≤lp0!⋯pl!Π0≤j≤m,0≤h≤lph,j!mmmmmmmmm·fp_Z0,…,ZpZ¯01,…,Z¯l1,X¯1·∂Z0p0,0⋯∂Zlpl,0emZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1·∏j=1m∂Z0p0,j⋯∂Zlpl,j∂X-SHZ0,…,Zl,Xmmmmnnnnnnnnnnni+wZ0,…,Zl,X∂X-SZ¯01,…,Z¯l1,X¯1.
By Lemma 3, one also gets a constant C4.1>0 (depending on p0,0,…,pl,0,X¯0,X¯1,Z¯j0,Z¯j1 for 0≤j≤l) with
(51)∂Z0p0,0⋯∂Zlpl,0emZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤C4.1Z¯01-p0,0⋯Z¯l1-pl,0emZ0,…,Zl,XZ¯00,…,Z¯l0,X¯0.
By Lemma 2, one finds
(52)∂Z0p0,j⋯∂Zlpl,j∂X-SHZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤Z¯01-p0,j⋯Z¯l1-pl,jX¯1SHZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤Z¯01-p0,j⋯Z¯l1-pl,jX¯1SR.
Due to Lemma 3, we get a constant C4.2>0 (depending on p0,j,…,pl,j,X¯0,X¯1,Z¯m0,Z¯m1 for 0≤m≤l) such that
(53)∂Z0p0,j⋯∂Zlpl,jwZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤∑q=0S-1∂Z0p0,j⋯∂Zlpl,jφqZ0,…,ZlZ¯01,…,Z¯l1,X¯1Xqq!Z¯01,…,Z¯l1,X¯1≤C4.2∑q=0S-1Z¯01-p0,j⋯Z¯l1-pl,jφqZ0,…,ZlZ¯00,…,Z¯l0,X¯0·Xqq!Z¯00,…,Z¯l0,X¯0≤C4.2∑q=0S-1Z¯01-p0,j⋯Z¯l1-pl,jδXqq!Z¯00,…,Z¯l0,X¯0.
Now, we can choose0<X¯1<X¯0, δ>0 and the constant K4>0 (recall that R=K4δ) in such a way that
(54)∑k_=(k0,…,kl+1)∈D1dk_Z0,…,Zl,X(Z¯00,…,Z¯l0,X¯0)Z¯l1-kl·Z¯01-k0⋯Z¯l1-klZ¯l1-kl·X¯1S-kl+1K4δ+∑q=0S-1Xqq!(Z¯00,…,Z¯l0,X¯0)C4δ+∑p_=(p0,…,pl)∈D2∑m∈D3∑∑j=0mph,j=ph,0≤h≤lp0!⋯pl!Π0≤j≤m,0≤h≤lph,j!·fp_Z0,…,Zl(Z¯00,…,Z¯l0,X¯0)×C4.1Z¯01-p0,0⋯Z¯l1-pl,0·emZ0,…,Zl,X(Z¯00,…,Z¯l0,X¯0)·∏j=1mZ¯01-p0,j⋯Z¯l1-pl,jX¯1SK4δ∑q=0S-1Z¯01-p0,j⋯Z¯l1-pl,jδXqq!Z¯00,…,Z¯l0,X¯0mmmi+C4.2∑q=0S-1Z¯01-p0,j⋯Z¯l1-pl,jδXqq!Z¯00,…,Z¯l0,X¯0≤K4δ.
Hence, gathering (48), (49), (50), (51), (52), (53), and (54) yields inclusion (46).
We turn to the proof of (ii). As above, we fixZ¯j1>0, 0≤j≤l, be real numbers such that M0<Z¯j1<Zj0. We also consider δ>0 for which (42) holds. Let R>0 be of the form R=K4δ for some constant K4>0. We take H1,H2∈BR where BR is the closed ball of radius R, centered at 0 in G(Z¯01,…,Z¯l1,X¯1) for some real number 0<X¯1<X¯0. From Lemmas 2 and 4, we get that
(55)dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1-Sj·H2Z0,…,Zl,X-H1Z0,…,Zl,Xdk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1-SZ¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1·Z¯01-k0⋯Z¯l1-klX¯1S-kl+1·H2Z0,…,Zl,X-H1Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯00,…,Z¯l0,X¯0·Z¯01-k0⋯Z¯l1-klX¯1S-kl+1·H2Z0,…,Zl,X-H1Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1.
Using the identity bm-am=(b-a)∑s=0m-1‍asbm-1-s for any complex numbers a,b and any integer m≥2, we can write
(56)∂X-SH1Z0,…,Zl,X+wZ0,…,Zl,Xmji-∂X-SH2Z0,…,Zl,X+wZ0,…,Zl,Xm=∂X-SH1-∂X-SH2=·∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-s.
Using Leibniz formula, we deduce from Lemma 4 that
(57)fp_Z0,…,Zl∂Z0p0⋯∂Zlplem∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-sm·Z0,…,Zl,X∂X-SH1-∂X-SH2m·∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-sZ¯01,…,Z¯l1,X¯1≤∑∑j=02ph,j=ph,0≤h≤lp0!⋯pl!Π0≤j≤2,0≤h≤lph,j!fp_Z0,…,ZlZ¯01,…,Z¯l1,X¯1·∂Z0p0,0⋯∂Zlpl,0emZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1·∂Z0p0,1⋯∂Zlpl,1∂X-SH1-∂X-SH2Z¯01,…,Z¯l1,X¯1·∂Z0p0,2⋯∂Zlpl,2∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-smii·∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-sZ¯01,…,Z¯l1,X¯1.
By Lemma 3, one also gets a constant C4.3>0 (depending on p0,0,…,pl,0,X¯0,X¯1,Z¯j0,Z¯j1 for 0≤j≤l) with
(58)∂Z0p0,0⋯∂Zlpl,0emZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤C4.3Z¯01-p0,0⋯Z¯l1-pl,0emZ0,…,Zl,XZ¯00,…,Z¯l0,X¯0.
By Lemma 2, one finds
(59)∂Z0p0,1⋯∂Zlpl,1∂X-SH1-∂X-SH2Z¯01,…,Z¯l1,X¯1≤Z¯01-p0,1⋯Z¯l1-pl,1X¯1SH1-H2Z¯01,…,Z¯l1,X¯1.
Using again Leibniz formula and Lemma 4, we can write
(60)∂Z0p0,2⋯∂Zlpl,2∂X-SH2+ws∂X-SH1+wm-1-sZ¯01,…,Z¯l1,X¯1≤∑∑j=1m-1ph,2,j=ph,2,0≤h≤lp0,2!⋯pl,2!Π1≤j≤m-1,0≤h≤lph,2,j!·∏j=1s∂Z0p0,2,j⋯∂Zlpl,0,j∂X-SH2+wZ¯01,…,Z¯l1,X¯1·∏j=s+1m-1∂Z0p0,2,j⋯∂Zlpl,0,j∂X-SH1+wZ¯01,…,Z¯l1,X¯1.
By Lemmas 2 and 3, one finds a constant C4.4>0 (depending on p0,2,j,…,pl,2,j,X¯0,X¯1,Z¯j0,Z¯j1 for 0≤j≤l) such that
(61)∂Z0p0,2,j⋯∂Zlpl,2,j∂X-SHr+wZ¯01,…,Z¯l1,X¯1≤Z¯01-p0,2,j⋯Z¯l1-pl,2,jX¯1SR+C4.4∑q=0S-1Z¯01-p0,2,j⋯Z¯l1-pl,2,jδXqq!Z¯00,…,Z¯l0,X¯0
for r=1,2.
In the following, we choose0<X¯1<X¯0 in order that
(62)∑k_=(k0,…,kl+1)∈D1dk_Z0,…,Zl,XZ¯00,…,Z¯l0,X¯0mmmmmmm·Z¯01-k0⋯Z¯l1-klX¯1S-kl+1+∑p_=(p0,…,pl)∈D2∑m∈D3∑∑j=02ph,j=ph,0≤h≤lp0!⋯pl!Π0≤j≤2,0≤h≤lph,j!mmmmmmmmmmmmimmm·fp_Z0,…,ZlZ¯00,…,Z¯l0,X¯0·C4.3Z¯01-p0,0⋯Z¯l1-pl,0emZ0,…,Zl,XZ¯00,…,Z¯l0,X¯0·Z¯01-p0,1⋯Z¯l1-pl,1X¯1S·∑s=0m-1∑∑j=1m-1ph,2,j=ph,2,0≤h≤lp0,2!⋯pl,2!Π1≤j≤m-1,0≤h≤lph,2,j!mm·∏j=1sZ¯01-p0,2,j⋯Z¯l1-pl,2,jX¯1SK4δδXqq!Z¯00,…,Z¯l0,X¯0mmmmmmi+C4.4∑q=0S-1Z¯01-p0,2,j⋯Z¯l1-pl,2,jmmmmmmmmmmmm·δXqq!Z¯00,…,Z¯l0,X¯0mm·∏j=s+1m-1Z¯01-p0,2,j⋯Z¯l1-pl,2,jX¯1SK4δδXqq!Z¯00,…,Z¯l0,X¯0mmmmmmmi+C4.4∑q=0S-1Z¯01-p0,2,j⋯Z¯l1-pl,2,jmmmmmmmmmmmmm·δXqq!Z¯00,…,Z¯l0,X¯0∑s=0m-1∑∑j=1m-1ph,2,j=ph,2,0≤h≤lp0,2!⋯pl,2!Π1≤j≤m-1,0≤h≤lph,2,j!≤12.
Taking into account all inequalities (55), (56), (57), (58), (59), (60), and (61) under constraint (62), we deduce (47).
Finally, for fixed real numbersZ¯j1>0, 0≤j≤l, such that M0<Z¯j1<Zj0, we choose 0<X¯1<X¯0, δ>0 and the constant K4>0 (recall that R=K4δ) in such a way that both constraints (54) and (62) hold. For these constants, the map A satisfies both (46) and (47).We are in a position to give the proof of Proposition5. Let Z¯j1>0, 0≤j≤l, be real numbers such that M0<Z¯j1<Zj0; we choose 0<X¯1<X¯0, δ>0 and the constant K4>0 as in Lemma 6. We have put R=δK4.From the fact thatG(Z¯01,…,Z¯l1,X¯1) equipped with the norm ·(Z¯01,…,Z¯l1,X¯1) is a Banach space, the closed ball (BR,d) for the metric d(x,y)=y-x(Z¯01,…,Z¯l1,X¯1) is a complete metric space. From Lemma 6, the map A is a contraction from (BR,d) into itself. From the classical fixed point theorem, we deduce that there exists a unique Hf(Z0,…,Zl,X)∈BR such that A(Hf)=Hf.By construction and taking into account Lemma2, the formal series
(63)UfZ0,…,Zl,X=∂X-SHfZ0,…,Zl,X+wZ0,…,Zl,X
is a solution of the problem (40), (41). Moreover, Uf(Z0,…,Zl,X)∈G(Z¯01,…,Z¯l1,X¯1) and
(64)UfZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤δX¯1SK4+∑j=0S-1Xjj!Z¯00,…,Z¯l0,X¯0
which yields (43). Thus proof of Proposition 5 is complete.


### 2.2. Weighted Banach Spaces of Holomorphic Functions on Sectors

We denote byD(0,r) the open disc centered at 0 with radius r>0 in C. Let ϵ0>0 be a real number and let Sd be an unbounded sector in direction d∈R centered at 0 in C. By convention, these sectors do not contain the origin in C. For any open set D⊂C, we denote by O(D) the vector space of holomorphic functions on D. Let l≥1 be an integer. For all tuples β_=(β0,…,βl+1)∈Nl+2, let ρβ_≥0 be positive real numbers such that ρβ_′≤ρβ_ if |β_′|≥|β_| (where by definition |β_|=∑j=0l+1‍βj). We define Ωβ_=D(0,ρβ_)∪Sd.Definition 7.
Letb>1 be a real number and let rb(β_)=∑n=0|β_|‍1/(n+1)b for all tuples β_∈Nl+2. Let ϵ∈D(0,ϵ0)∖{0} and r,σ>0 be real numbers. We denote by Eβ_,ϵ,σ,r,Ωβ_ the vector space of all functions v∈O(Ωβ_) such that
(65)vτβ_,ϵ,σ,r,Ωβ_≔sup⁡τ∈Ωβ_vτ1+τ2ϵ2rexp⁡-σϵrrbβ_τ
is finite.Remark 8.
These norms are appropriate modifications of the norms defined in [4, 17].
In the next proposition, we study some parameter depending linear operators acting on the spacesEβ_,ϵ,σ,r,Ωβ_.Proposition 9.
Lets1,k0≥0 be integers. Let β_=(β0,…,βl+1)∈Nl+2, β~_=(β~0,…,β~l+1)∈Nl+2. Under the assumption that |β~_|>|β_|, the operator τs1∂τ-k0 defines a bounded linear map from Eβ_,ϵ,σ,r,Ωβ_ into Eβ~_,ϵ,σ,r,Ωβ~_, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, one has
(66)τs1∂τ-k0Vβ_τ,ϵβ~_,ϵ,σ,r,Ωβ~_≤Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_·ϵrs1+k0β~_+1bs1+k0s1+k0e-1σβ~_-β_s1+k0mmmmmmm+β~_+1bs1+k0+2mmmmmmm·s1+k0+2e-1σβ~_-β_s1+k0+2
for all Vβ_(τ,ϵ)∈Eβ_,ϵ,σ,r,Ωβ_.Proof.
The proof follows the same lines of arguments as Lemma 1 from [17]. By construction of the operator ∂τ-k0, one can write
(67)τs1∂τ-k0Vβ_τ,ϵ=τs1+k0∫01⋯∫01Vβ_hk0⋯h1τ,ϵ=τs1+k0n∫01⋯u·1+hk0⋯h1τ2ϵ2r=τs1+k0n∫01⋯u·exp⁡-σϵrrbβ_hk0⋯h1τ=τs1+k0n∫01⋯u·exp⁡σ/ϵrrbβ_hk0⋯h1τ1+hk0⋯h1τ2/ϵ2r=τs1+k0n∫01⋯u·Mk0h1,…,hk0-1dhk0⋯dh1,
where Mk0(h1,…,hk0-1) is a monic monomial for k0≥2, while M1=1, for all τ∈Ωβ_. We deduce that
(68)τs1∂τ-k0Vβ_τ,ϵ1+τ2ϵ2rexp⁡-σϵrrbβ~_τ≤τs1+k0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_1+τ2ϵ2r·exp⁡-σϵrrbβ~_-rbβ_τ
for all τ∈Ωβ~_⊂Ωβ_. By definition of rb, one has
(69)rbβ~_-rbβ_=∑n=|β_|+1|β~_|1n+1b≥β~_-β_β~_+1b.
Now, we recall the classical estimates: for all integers m1,m2≥0,
(70)sup⁡x≥0xm1e-m2x=m1m2m1e-m1
holds. Using (70), we deduce that
(71)τs1+k0+τs1+k0+2ϵ2rexp⁡-σϵrβ~_-β_β~_+1bτ≤ϵrs1+k0β~_+1bs1+k0s1+k0e-1σβ~_-β_s1+k0mmmmmmmmm+β~_+1bs1+k0+2mmmmmmmmm·s1+k0+2e-1σβ~_-β_s1+k0+2.
Taking into account (68), (69), and (71), we get (66).In the next proposition, we study the convolution product of functions in the spacesEβ_,ϵ,σ,r,Ωβ_.Proposition 10.
Letβ_1=(β01,…,βl+11)∈Nl+2 and β_2=(β02,…,βl+12)∈Nl+2. Let β_∈Nl+2 such that |β_|≥|β_1|+|β_2|. Then, for any Vβ_1(τ,ϵ)∈Eβ_1,ϵ,σ,r,Ωβ_1, Vβ_2(τ,ϵ)∈Eβ_2,ϵ,σ,r,Ωβ_2, the convolution product Vβ_1(τ,ϵ)*Vβ_2(τ,ϵ) belongs to Eβ_,ϵ,σ,r,Ωβ_, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, there exists a universal constant C5>0 such that
(72)∫0τVβ_1τ-s,ϵVβ_2s,ϵdsβ_,ϵ,σ,r,Ωβ_≤C5ϵrVβ_1τ,ϵβ_1,ϵ,σ,r,Ωβ_1Vβ_2τ,ϵβ_2,ϵ,σ,r,Ωβ_2
for all Vβ_1(τ,ϵ)∈Eβ_1,ϵ,σ,r,Ωβ_1, Vβ_2(τ,ϵ)∈Eβ_2,ϵ,σ,r,Ωβ_2.Proof.
We mimic the proof of Lemma 3 in [17]. One can write
(73)∫0τVβ_1τ-s,ϵVβ_2s,ϵds=∫0τVβ_1τ-s,ϵ1+τ-s2ϵ2rmmii·exp⁡-σϵrrbβ_1τ-smmii·Vβ_2s,ϵ1+s2ϵ2rexp⁡-σϵrrbβ_2smmii·exp⁡σ/ϵrrbβ_1τ-s+rbβ_2s1+τ-s2/ϵ2r1+s2/ϵ2rds
for all τ∈Ωβ_⊂Ωβ_1∩Ωβ_2. We deduce that
(74)∫0τVβ_1τ-s,ϵVβ_2s,ϵds≤Vβ_1τ,ϵβ_1,ϵ,σ,r,Ωβ_1Vβ_2τ,ϵβ_2,ϵ,σ,r,Ωβ_2·∫01τexp⁡⁡στ/ϵrrbβ_11-h+rbβ_2h1+τ2/ϵ2r1-h21+τ2/ϵ2rh2dh
for all τ∈Ωβ_. Since rb is increasing, one has
(75)rbβ_11-h+rbβ_2h≤rbβ_.
Therefore,
(76)1+τ2ϵ2rexp⁡⁡-σϵrrbβ_τ·∫01τexp⁡⁡στ/ϵrrbβ_11-h+rbβ_2h1+τ2/ϵ2r1-h21+τ2/ϵ2rh2dh≤∫011+τ2/ϵ2rτ1+τ2/ϵ2r1-h21+τ2/ϵ2rh2dh=Jτ,ϵ
for all τ∈Ωβ_. Now, from [28], we know that there exists a universal constant C5>0 such that
(77)Jϵrτ,ϵϵr≤C5
for all τ∈C, ϵ∈C*. We deduce that
(78)sup⁡τ∈CJτ,ϵϵr=sup⁡τ∈CJϵrτ,ϵϵr≤C5
for all ϵ∈C*. Finally, gathering (74), (76), and (78) yields estimates (72).Corollary 11.
Letl0≥0 be an integer. The operator ∂τ-l0 defines a bounded linear map from Eβ_,ϵ,σ,r,Ωβ_ into itself, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, there exists a constant C6>0 (depending on l0,σ) such that
(79)∂τ-l0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_≤C6ϵrl0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_
for all Vβ_(τ,ϵ)∈Eβ_,ϵ,σ,r,Ωβ_.Proof.
We carry out a similar proof as in Corollary 1 from [17]. We denote by χC the function equal to 1 on C. By definition, we put χC*1=χC and χC*l means the convolution product of χC, l-1 times for l≥2. By definition, we can write ∂τ-l0Vβ_(τ,ϵ)=(χC(τ))*l0*Vβ_(τ,ϵ). From Proposition 10, there exists a (universal) constant C5>0 such that
(80)∂τ-l0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_≤C5l0ϵrl0χCτ0_,ϵ,σ,r,Ω0_l0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_,
where 0_=(0,…,0)∈Nl+2. By Definition 7 and using formula (70), we have that
(81)χCτ0_,ϵ,σ,r,Ω0_=sup⁡τ∈Ω0_1+τ2ϵ2rexp⁡-σϵrτ≤1+2e-1σ2.
From estimates (80) and (81), we get inequality (79).The next proposition involves bound estimates for multiplication operators of bounded holomorphic functions.Proposition 12.
Leta(τ,ϵ) be a holomorphic function on Ω0_×D(0,ϵ0)∖{0} such that there exists a constant M>0 with sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)∖{0}|a(τ,ϵ)|≤M. Then, the multiplication by a(τ,ϵ) is a bounded linear operator from Eβ_,ϵ,σ,r,Ωβ_ into itself, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, the inequality
(82)aτ,ϵVβ_τ,ϵβ_,ϵ,σ,r,Ωβ_≤MVβ_τ,ϵβ_,ϵ,σ,r,Ωβ_
holds for all Vβ_(τ,ϵ)∈Eβ_,ϵ,σ,r,Ωβ_.Proof.
The proof is a direct consequence of Definition7 for the norm ·β_,ϵ,σ,r,Ωβ_.


### 2.3. A Global Cauchy Problem

We keep the same notations as in the previous section. In the following, we introduce some definitions. LetA1 be a finite subset of N3 and let A2 be a finite subset of N2. Let l≥1 be an integer.For all(k0,k1,k2)∈A1, we denote by I(k0,k1,k2) a finite subset of N2. For all (k0,k1,k2)∈A1, all (s1,s2)∈I(k0,k1,k2), and all integers β0,βl+1≥0, we denote by as1,s2,k0,k1,k2,β0,βl+1(τ,ϵ) some holomorphic function on Ω0_×D(0,ϵ0) which satisfies the estimates: there exist constants ρ,ρ′>0, as1,s2,k0,k1,k2>0 with
(83)sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)as1,s2,k0,k1,k2,β0,βl+1τ,ϵ≤as1,s2,k0,k1,k2e-ρ′2β012ρβl+1β0!βl+1!
for all (k0,k1,k2)∈A1, we consider the series
(84)a(k0,k1,k2)τ,z,x,ϵ=∑(s1,s2)∈I(k0,k1,k2)∑β0≥0,βl+1≥0as1,s2,k0,k1,k2,β0,βl+1mmmmmmmmmmmmmm·τ,ϵτs1ϵ-s2eizβ0β0!xβl+1βl+1!
which define holomorphic functions on Ω0_×Hρ′×D(0,ρ)×D(0,ϵ0)∖{0}, where Hρ′ is defined as the following strip in C:
(85)Hρ′=z∈CIm⁡z<ρ′.
For all (l0,l1)∈A2, we denote by J(l0,l1) a finite subset of N. For all m1∈J(l0,l1) and all integers β0,βl+1≥0, we denote by αm1,l0,l1,β0,βl+1(τ,ϵ) some bounded holomorphic function on Ω0_×D(0,ϵ0) with the following estimates: there exists a constant am1,l0,l1>0 such that
(86)sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)αm1,l0,l1,β0,βl+1τ,ϵ≤am1,l0,l1e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0.For all(l0,l1)∈A2, we consider the series
(87)α(l0,l1)τ,z,x,ϵ=∑m1∈J(l0,l1)∑β0≥0,βl+1≥0αm1,l0,l1,β0,βl+1τ,ϵϵ-m1eizβ0β0!xβl+1βl+1!
which define holomorphic functions on Ω0_×Hρ′×D(0,ρ)×D(0,ϵ0)∖{0}.Letl≥1 be an integer and let ξ0=1 and ξ1,…,ξl be real algebraic numbers such that the family {1,ξ1,…,ξl} is Z-linearly independent (this means that each ξj is a real root of a polynomial Pj∈Z[X] and if there exist integers k0,…,kl∈Z such that k0+∑j=1l‍kjξj=0, then kj=0 for 0≤j≤l). Due to the classical primitive element theorem of Artin, we consider an algebraic number field K=Q(ξ) containing the numbers ξj, 1≤j≤l and we denote by h+1≥1 its degree (that is the dimension of the vector space Q(ξ) over Q). The following lemma is a direct consequence of Theorem 11 in [29].Lemma 13.
There exists a constantCξ1,…,ξl>0 (depending on ξ1,…,ξl) such that, for any k_=(k0,…,kl)∈Zl+1∖{0}, the inequality
(88)k0+k1ξ1+⋯+klξl≥Cξ1,…,ξlmax⁡j=0lkjh≥Cξ1,…,ξlk0+⋯+klh
holds.Example 14.
Letξ be an algebraic number. Assume that the degree of Q(ξ) is h+1. Then, the algebraic numbers {1,ξ,…,ξh} are Z-linearly independent and inequality (88) above holds for ξj=ξj, 0≤j≤h. In that case, one recovers Lemma 2.1 of [6].
LetS,r1,r2≥1 be integers. We put
(89)ρβ_=Cξ1,…,ξlr1/r22β0+1+β1+⋯+βl+1hr1/r2
for all β_=(β0,…,βl+1)∈Nl+2. We consider an unbounded sector Sd⊂C centered at 0 such that
(90)argτ≠2k+1r2π,0≤k≤r2-1,
for all τ∈Sd. As in the previous section, we put Ωβ_=D(0,ρβ_)∪Sd. For all 0≤j≤S-1, we choose a set of functions V(β0,…,βl,j)(τ,ϵ)∈E(β0,…,βl,j),ϵ,σ,r,Ω(β0,…,βl,j) for all β0,…,βl≥0 and we consider the formal series
(91)Vjτ,z,ϵ=∑β0,…,βl≥0Vβ0,…,βl,jτ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!
for all 0≤j≤S-1.
We consider the following Cauchy problem:(92)τr2+-i∂z+1r1∂xSVτ,z,x,ϵ=∑(k0,k1,k2)∈A1ak0,k1,k2τ,z,x,ϵ∂τ-k0∂zk1∂xk2Vτ,z,x,ϵ+∑(l0,l1)∈A2,l1≥2αl0,l1τ,z,x,ϵ∂τ-l0Vτ,z,x,ϵ*l1,
where V*1=V and V*l1, l1≥2, stands for the convolution product of V applied l1-1 times with respect to τ, for given initial conditions
(93)∂zjVτ,z,0,ϵ=Vjτ,z,ϵ,0≤j≤S-1.
In the sequel, we will need the next lemma.Lemma 15.
There exists a constantC7>0 (depending on r1,r2,Cξ1,…,ξl,Sd) such that
(94)1τr2+1+∑j=0lβjξjr1≤C71+∑j=0lβjhr1
for all τ∈Ω(β0,…,βl,βl+1+S), for all βj≥0, 0≤j≤l+1.Proof.
We putA=1+∑j=0l‍βjξj. The following partial fraction decomposition holds:
(95)1τr2+Ar1=∑k=0r2-1Akτ-Ar1/r2eiπ2k+1/r2,
where
(96)Ak=1r2e-iπ2k+1r2-1/r2Ar1-r1/r2
for all 0≤k≤r2-1. Now, there exists some constant C8>0 (depending on Sd) such that
(97)τ-1+∑j=0lβjξjr1/r2eiπ2k+1/r2≥C81+∑j=0lβjξjr1/r2
for all τ∈Ω(β0,…,βl,βl+1+S). Indeed, from (88), we know that
(98)1+∑j=0lβjξjr1/r2≥Cξ1,…,ξlr1/r21+∑j=0lβjhr1/r2>Cξ1,…,ξlr1/r21+∑j=0l+1βj+Shr1/r2
for all βj≥0, 0≤j≤l+1. Let τ∈D(0,ρ(β0,…,βl,βl+1+S)). From (98), we can write
(99)τ=h2eiθ1+∑j=0lβjξjr1/r2
for some 0≤h<1 and θ∈[0,2π). Therefore,
(100)τ-1+∑j=0lβjξjr1/r2eiπ2k+1/r2=1+∑j=0lβjξjr1/r2h2eiθ-eiπ2k+1/r2≥121+∑j=0lβjξjr1/r2.
Now, let τ∈Sd. For all k∈{0,…,r2-1}, we can write
(101)τ=s1+∑j=0lβjξjr1/r2eiπ2k+1/r2+sk,
where s≥0 and sk∈R. By construction of Sd, we have that eisk≠1 for all k∈{0,…,r2-1}. As a result, there exists a constant C9>0 (depending on Sd) with |seisk-1|≥C9, for all s≥0. Hence,
(102)τ-1+∑j=0lβjξjr1/r2eiπ2k+1/r2=1+∑j=0lβjξjr1/r2seiπ2k+1/r2+sk-eiπ2k+1/r2≥C91+∑j=0lβjξjr1/r2.
As a consequence, we get that (97) holds. On the other hand, from (88), we get that
(103)1Ar1≤1+∑j=0lβjhr1Cξ1,…,ξlr1
for all βj≥0, 0≤j≤l. Gathering (95), (97), and (103), we deduce that
(104)Akτ-Ar1/r2eiπ2k+1/r2≤1r2C8Ar1≤1r2C8Cξ1,…,ξlr11+∑j=0lβjhr1
for all 0≤k≤r2-1. The lemma follows.In the next proposition, we construct formal series solutions of (92) and (93).Proposition 16.
Under the assumption that(105)S>k2
for all (k0,k1,k2)∈A1, there exists a formal series
(106)Vτ,z,x,ϵ=∑β_=(β0,…,βl+1)∈Nl+2Vβ_τ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!
solution of (92) and (93), where the coefficients τ↦Vβ_(τ,ϵ) belong to the space O(Ωβ_) for all β_∈Nl+2 and satisfy the following recursion formula:
(107)τr2+1+∑j=0lβjξjr1Vβ0,…,βl,βl+1+Sτ,ϵβ0!⋯βl!βl+1!=∑(k0,k1,k2)∈A1∑(s1,s2)∈Ik0,k1,k2∑β01+β02=β0βl+11+βl+12=βl+1as1,s2,k0,k1,k2,β01,βl+11τ,ϵβ01!βl+11!+∑l0,l1∈A2,l1≥2∑m∈Jl0,l1hhhhhhhhhhhhhh×τs1ϵ-s2∂τ-k0Vβ02,β1,…,βl,βl+12+k2τ,ϵβ02!β1!⋯βl!βl+12!ik1β02+∑j=1lβjξjk1+∑(l0,l1)∈A2,l1≥2∑m∈J(l0,l1)∑β0-1+β00+⋯+β0l1-1=β0βj0+⋯+βjl1-1=βj,1≤j≤lβl+1-1+βl+10+⋯+βl+1l1-1=βl+1αm1,l0,l1,β0-1,βl+1-1τ,ϵϵ-m1β0-1!βl+1-1!hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh×∂τ-l0Vβ00,…,βl+10τ,ϵ*⋯*Vβ0l1-1,…,βl+1l1-1τ,ϵΠm=0l1-1Πj=0l+1βjm!
for all β0,…,βl+1≥0, all τ∈Ω(β0,…,βl,βl+1+S), and all ϵ∈D(0,ϵ0)∖{0}.Proof.
By hypothesis, we know in particular thatV(β0,…,βl,j)(τ,ϵ) belongs to O(Ω(β0,…,βl,j)) for any β0,…,βl≥0, all 0≤j≤S-1, and all ϵ∈D0,ϵ0∖{0}. Since Ωβ_′⊂Ωβ_ when |β_′|>|β_| and using Lemma 15, one gets that the functions τ↦Vβ_(τ,ϵ) which are defined by recursion (107) actually belong to O(Ωβ_) for any β_∈Nl+2, for all ϵ∈D(0,ϵ0)∖{0}. Direct computation by identification of the powers of eiz,…,eizξl and the powers of x shows that the formal series (106) is solution of (92) and (93), if its coefficients Vβ_(τ,ϵ) satisfy recursion (107).In the next proposition, we state norm inequalities for the sequenceVβ_.Proposition 17.
We consider the sequence of functionsVβ_(τ,ϵ) defined by recursion (107) for given initial data V(β0,…,βl,j)(τ,ϵ) defined above for all β0,…,βl≥0, 0≤j≤S-1. Then, for all β_∈Nl+2 and all ϵ∈D(0,ϵ0)∖{0}, the function τ↦Vβ_(τ,ϵ) belongs to Eβ_,ϵ,σ,r,Ωβ_. We put wβ_(ϵ)=Vβ_(τ,ϵ)β_,ϵ,σ,r,Ωβ_, for all β_∈Nl+2 and all ϵ∈D(0,ϵ0)∖{0}. Then, the sequence wβ_(ϵ) satisfies the following estimates. There exist constants C10>0 (depending on r1,r2,Cξ1,…,ξl,Sd) and C11>0 (depending on σ) such that
(108)wβ0,…,βl,βl+1+Sϵβ0!⋯βl!βl+1!≤∑(k0,k1,k2)∈A1∑(s1,s2)∈I(k0,k1,k2)C101+∑j=0lβjhr1·∑β01+β02=β0βl+11+βl+12=βl+1As1,s2,k0,k1,k2,β01,βl+11β01!βl+11!mmmmm·wβ02,β1,…,βl,βl+12+k2ϵβ02!β1!⋯βl!βl+12!ϵrs1+k0-s2·∑j=0l+1βj+Sbs1+k0s1+k0e-1σS-k2s1+k0mi+∑j=0l+1βj+Sbs1+k0+2s1+k0+2e-1σS-k2s1+k0+2·β0+∑j=1lβjξjk1+∑(l0,l1)∈A2,l1≥2∑m1∈J(l0,l1)C101+∑j=0lβjhr1·∑β0-1+β00+⋯+β0l1-1=β0βj0+⋯+βjl1-1=βj,1≤j≤lβl+1-1+βl+10+⋯+βl+1l1-1=βl+1Bm1,l0,l1,β0-1,βl+1-1β0-1!βl+1-1!mmmmmmmm·C11l1ϵrl0+l1-1-m1Πj=0l1-1wβ0j,…,βl+1jϵΠj=0l1-1β0j!⋯βl+1j!
for all βj≥0, 0≤j≤l+1, and all ϵ∈D(0,ϵ0)∖{0}, where
(109)As1,s2,k0,k1,k2,β01,βl+11=sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)∖0as1,s2,k0,k1,k2,β01,βl+11τ,ϵ,Bm1,l0,l1,β0-1,βl+1-1=sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)∖0αm1,l0,l1,β0-1,βl+1-1τ,ϵ.Proof.
We apply the norm·(β0,…,βl,βl+1+S),ϵ,σ,r,Ω(β0,…,βl,βl+1+S) on the left and right hand side of equality (107) and use Propositions 9, 10, and 12, Lemma 15, and Corollary 11 in order to majorize the right hand side. Indeed, using Propositions 9, 10, and 12, Corollary 11, Lemma 15, and the estimates
(110)β02+∑j=1lβjξjk1≤β0+∑j=1lβjξjk1
we get a constant C7>0 (depending on r1,r2,Cξ1,…,ξl,Sd) such that
(111)1τr2+1+∑j=0lβjξjr1as1,s2,k0,k1,k2,β01,βl+11τ,ϵβ01!βl+11!m·τs1ϵ-s2×∂τ-k0Vβ02,β1,…,βl,βl+12+k2τ,ϵβ02!β1!⋯βl!βl+12!m·ik1β02+∑j=1lβjξjk1β0,…,βl,βl+1+S≤C71+∑j=0lβjhr1×As1,s2,k0,k1,k2,β01,βl+11β01!βl+11!·wβ02,β1,…,βl,βl+12+k2ϵβ02!β1!⋯βl!βl+12!ϵrs1+k0-s2·∑j=0l+1βj+Sbs1+k0s1+k0e-1σS-k2s1+k0mi+∑j=0l+1βj+Sb(s1+k0+2)s1+k0+2e-1σS-k2s1+k0+2·β0+∑j=1lβjξjk1
and using Propositions 10 and 12 and Corollary 11, we obtain a universal constant C5>0 and some constants C6>0 (depending on l0,σ) and C7>0 (depending on r1,r2,Cξ1,…,ξl,Sd) such that
(112)1τr2+1+∑j=0lβjξjr1αm1,l0,l1,β0-1,βl+1-1τ,ϵϵ-m1β0-1!βl+1-1!∂τ-l0Vβ00,…,βl+10τ,ϵ*⋯*Vβ0l1-1,…,βl+1l1-1τ,ϵΠm=0l1-1Πj=0l+1βjm!i·∂τ-l0Vβ00,…,βl+10τ,ϵ*⋯*Vβ0l1-1,…,βl+1l1-1τ,ϵΠm=0l1-1Πj=0l+1βjm!β0,…,βl,βl+1+S≤C71+∑j=0lβjhr1Bm1,l0,l1,β0-1,βl+1-1β0-1!βl+1-1!·C6C5l1-1ϵrl0+l1-1-m1Πj=0l1-1wβ0j,…,βl+1jϵΠj=0l1-1β0j!⋯βl+1j!.We define the following formal series:(113)As1,s2,k0,k1,k2Z0,X=∑β0,βl+1≥0As1,s2,k0,k1,k2,β0,βl+1Z0β0β0!Xβl+1βl+1!,Bm1,l0,l1Z0,X=∑β0,βl+1≥0Bm1,l0,l1,β0,βl+1Z0β0β0!Xβl+1βl+1!.
We consider the following Cauchy problem:
(114)∂XSUZ0,…,Zl,X,ϵ=∑(k0,k1,k2)∈A1∑(s1,s2)∈I(k0,k1,k2)C101+∑j=0lZj∂Zjhr1·s1+k0e-1σS-k2s1+k0∑j=0lZj∂Zj+X∂X+Sb(s1+k0)mii+s1+k0+2e-1σS-k2s1+k0+2mii·∑j=0lZj∂Zj+X∂X+Sb(s1+k0+2)·∑j=0lξjZj∂Zjk1·ϵ0rs1+k0-s2As1,s2,k0,k1,k2Z0,X∂Xk2UZ0,…,Zl,X,ϵ+∑(l0,l1)∈A2,l1≥2∑m1∈J(l0,l1)C101+∑j=0lZj∂Zjhr1·ϵ0r(l0+l1-1)-m1C11l1Bm1,l0,l1Z0,XUZ0,…,Zl,X,ϵl1
for given initial data
(115)∂XjUZ0,…,Zl,0,ϵ=∑β0≥0,…,βl≥0wβ0,…,βl,jϵZ0β0⋯Zlβlβ0!⋯βl!,0≤j≤S-1
for all ϵ∈D(0,ϵ0)∖{0}.Proposition 18.
Under the assumption that(116)S>k2+bs1+k0+2
for all (k0,k1,k2)∈A1 and all (s1,s2)∈I(k0,k1,k2), there exists a formal series
(117)UZ0,…,Zl,X,ϵ=∑β_=(β0,…,βl+1)∈Nl+2Uβ_ϵZ0β0⋯ZlβlXβl+1β0!⋯βl!βl+1!
solution of (114) and (115), where the coefficients Uβ_(ϵ) satisfy the following recursion:
(118)Uβ0,…,βl,βl+1+Sϵβ0!⋯βl!βl+1!=∑(k0,k1,k2)∈A1∑(s1,s2)∈Ik0,k1,k2C101+∑j=0lβjhr1·∑β01+β02=β0βl+11+βl+12=βl+1As1,s2,k0,k1,k2,β01,βl+11β01!βl+11!mmmmm·Uβ02,β1,…,βl,βl+12+k2ϵβ02!β1!⋯βl!βl+12!ϵ0rs1+k0-s2·∑j=0l+1βj+Sb(s1+k0)s1+k0e-1σS-k2s1+k0mi+∑j=0l+1βj+Sbs1+k0+2mi·s1+k0+2e-1σS-k2s1+k0+2·β0+∑j=1lβjξjk1+∑(l0,l1)∈A2,l1≥2∑m1∈Jl0,l1C101+∑j=0lβjhr1·∑β0-1+β00+⋯+β0l1-1=β0βj0+⋯+βjl1-1=βj,1≤j≤lβl+1-1+βl+10+⋯+βl+1l1-1=βl+1Bm1,l0,l1,β0-1,βl+1-1β0-1!βl+1-1!·C11l1ϵ0r(l0+l1-1)-m1Πj=0l1-1Uβ0j,…,βl+1jϵΠj=0l1-1β0j!⋯βl+1j!
for all βj≥0, 0≤j≤l+1, and all ϵ∈D(0,ϵ0)∖{0}.Proposition 19.
Under assumption (116) with the additional condition that
(119)rs1+k0≥s2,rl0+l1-1≥m1,l1≥2,
for all (k0,k1,k2)∈A1, all (s1,s2)∈I(k0,k1,k2), and all (l0,l1)∈A2, l1≥2 and m1∈Jl0,l1, the following inequalities hold:
(120)wβ_ϵ≤Uβ_ϵ
for all β_∈Nl+2, all ϵ∈D(0,ϵ0)∖{0}.Proof.
By assumption (115), we know that
(121)Uβ0,…,βl,jϵ=wβ0,…,βl,jϵ
for all 0≤j≤S-1, all (β0,…,βl)∈Nl+1, and all ϵ∈D(0,ϵ0)∖{0}. Therefore, we get our result by using induction from inequalities (108) and equalities (118).In the next proposition we give sufficient conditions for the formal series solutions of the Cauchy problem (92), (93) in order to define actual holomorphic functions with exponential bound estimates.Proposition 20.
We make the assumption that (119) holds. We also assume that
(122)S>bs1+k0+2+k2,S≥hr1+bs1+k0+2+k1+k2
for all (k0,k1,k2)∈A1 and all (s1,s2)∈I(k0,k1,k2).
We choose two real numbersρ1′, M0>0 such that
(123)M0>2l+2exp⁡ρ1′max⁡j=0lξj
and we take X¯0>0 and Z¯j0>M0, for 0≤j≤l. We assume that the formal series
(124)φjZ0,…,Zl,ϵ=∑β0,…,βl≥0wβ0,…,βl,jϵZ0β0⋯Zlβlβ0!⋯βl!,0≤j≤S-1,
belong to G(Z¯00,…,Z¯l0,X¯0) for all ϵ∈D(0,ϵ0)∖{0} and that there exists a constant Cφj such that sup⁡ϵ∈D(0,ϵ0)∖{0}φj(Z¯00,…,Z¯l0,X¯0)≤Cφj. As a consequence, the formal series (91) define holomorphic functions Vj(τ,z,ϵ) on the product Sd×Hρ1′×D(0,ϵ0)∖{0} and satisfy the following estimates: there exists a constant C12>0 (depending on Cφj,l) such that
(125)Vjτ,z,ϵ≤C121+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,ϵ)∈Sd×Hρ1′×D(0,ϵ0)∖{0}, and all 0≤j≤S-1.
Then, there existsδ>0 (depending on M0, Z¯j0, |ξj| for 0≤j≤l, X¯0,r1,r2,Cξ1,…,ξl,Sd, h,σ, b,S, As1,s2,k0,k1,k2(Z0,X) for (k0,k1,k2)∈A1 and (s1,s2)∈I(k0,k1,k2) and Bm1,l0,l1(Z0,X) for (l0,l1)∈A2, m1∈J(l0,l1)) such that if one assumes moreover that
(126)φjZ0,…,Zl,ϵZ¯00,…,Z¯l0,X¯0<δ,0≤j≤S-1,
for all ϵ∈D(0,ϵ0)∖{0}, the formal series (106), solution of (92) and (93), defines a holomorphic function V(τ,z,x,ϵ) on the product Sd×Hρ1′×D(0,ρ1)×D(0,ϵ0)∖{0} for some ρ1>0 and carries the next bound estimates: there exists a constant C13>0 (depending on the same as for δ given above) with
(127)Vτ,z,x,ϵ≤C131+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,x,ϵ)∈Sd×Hρ1′×D(0,ρ1)×D(0,ϵ0)∖{0}.Proof.
Sinceφj(Z0,…,Zl,ϵ) belongs to G(Z¯00,…,Z¯l0,X¯0), we get a constant C14>0 (depending on Cφj) such that
(128)Vβ0,…,βl,jτ,ϵβ0,…,βl,j,ϵ,σ,r,Ωβ0,…,βl,j≤C141Z¯00β0⋯1Z¯l0βl∑q=0lβq!
for all β0,…,βl≥0 and all 0≤j≤S-1. From the multinomial formula, we know that
(129)∑q=0lβq!≤l+1∑q=0lβqβ0!⋯βl!
for all β0,…,βl≥0. Therefore, from (128), we deduce that
(130)Vβ0,…,βl,jτ,ϵ≤C141+τ2ϵ2r-1·exp⁡σϵrrb∑q=0lβq+jτ·l+1M0∑q=0lβqβ0!⋯βl!
for all β0,…,βl≥0, 0≤j≤S-1, all τ∈Ω(β0,…,βl,j), and all ϵ∈D(0,ϵ0)∖{0}. From assumption (123), we deduce that the formal series (91) defines a holomorphic function Vj(τ,z,ϵ) on the product Sd×Hρ1′×D(0,ϵ0)∖{0} and satisfies
(131)Vjτ,z,ϵ≤C141+τ2ϵ2r-1exp⁡σϵrζbτ·∑β0,…,βl≥0l+1exp⁡⁡ρ1′max⁡q=0l⁡ξqM0∑q=0lβq≤2l+1C141+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,ϵ)∈Sd×Hρ1′×D(0,ϵ0)∖{0} and all 0≤j≤S-1.
Under assumptions (119) and (122) together with (126), we see that the hypotheses of Proposition 5 are fulfilled for the Cauchy problem (114), (115). Therefore, we deduce that the formal solution U(Z0,…,Zl,X,ϵ) of (114), (115) constructed in Proposition 18 belongs to the space G(Z¯01,…,Z¯l1,X¯1) for some 0<X¯1<X¯0 and for some Z¯j1>M0. Moreover, we get a constant C15>0 (depending on Z¯j0,Z¯j1,|ξj| for 0≤j≤l, X¯0,r1,r2,Cξ1,…,ξl,Sd, h,σ,b,S, As1,s2,k0,k1,k2(Z0,X) for (k0,k1,k2)∈A1 and (s1,s2)∈I(k0,k1,k2) and Bm1,l0,l1(Z0,X) for (l0,l1)∈A2, m1∈J(l0,l1)) such that
(132)UZ0,…,Zl,X,ϵZ¯01,…,Z¯l1≤δC15
for all ϵ∈D(0,ϵ0)∖{0}. In particular, we deduce that
(133)Uβ0,…,βl+1ϵ≤δC151Z¯01β0⋯1Z¯l1βl1X¯1βl+1∑j=0l+1βj!
for all β0,…,βl+1≥0. Gathering (120) and (133) yields
(134)Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_≤δC151Z¯01β0⋯1Z¯l1βl1X¯1βl+1∑j=0l+1βj!
for all β_=(β0,…,βl+1)∈Nl+2. Again by the multinomial formula, we have that
(135)∑j=0l+1βj!≤l+2∑j=0l+1βjβ0!⋯βl+1!
for all β0,…,βl+1≥0. Hence, from (134), we get that
(136)Vβ_τ,ϵ≤δC151+τ2ϵ2r-1exp⁡σϵrrbβ_τ·l+2M0∑j=0lβjl+2X¯1βl+1β0!⋯βl+1!
for all β0,…,βl+1≥0, all τ∈Ωβ_, and all ϵ∈D(0,ϵ0)∖{0}. We deduce that the formal series (106) defines a holomorphic function V(τ,z,x,ϵ) on the product Sd×Hρ1′×D(0,X¯1/2(l+2))×D(0,ϵ0)∖{0} and satisfies
(137)Vτ,z,x,ϵ≤δC151+τ2ϵ2r-1exp⁡σϵrζbτ·∑β0,…,βl+1≥0l+2exp⁡⁡ρ1′max⁡j=0lξjM0∑j=0lβj·l+2xX¯1βl+1≤2l+2δC151+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,x,ϵ)∈Sd×Hρ1′×D(0,X¯1/2(l+2))×D(0,ϵ0)∖{0}.


## 2.1. A Cauchy-Kowalevski Theorem in Several Variables

In this section, we recall the well-known Cauchy-Kowalevski theorem in some spaces of analytic functions for which the size of the domain of existence of the solution can be controlled in terms of some supremum norm of the initial data.The next Banach spaces are natural extensions to the several variables cases of the spaces used in [27].Definition 1.
Letl≥1 be some integer. Let Z¯0,…,Z¯l,X¯>0 be positive real numbers. We denote by G(Z¯0,…,Z¯l,X¯) the space of formal series
(23)UZ0,…,Zl,X=∑β_=(β0,…,βl,βl+1)∈Nl+2uβ_Z0β0⋯ZlβlXβl+1β0!⋯βl!βl+1!
that belong to C[[Z0,…,Zl,X]] such that
(24)UZ0,…,Zl,XZ¯0,…,Z¯l,X¯=∑β_=(β0,…,βl,βl+1)∈Nl+2uβ_Z¯0β0⋯Z¯lβlX¯βl+1∑j=0l+1βj!
is finite. One can show that G(Z¯0,…,Z¯l,X¯) equipped with the norm ·(Z¯0,…,Z¯l,X¯) are Banach spaces.In the next two lemmas we show continuity properties for some linear integrodifferential operators acting on the aforementioned Banach spaces.Lemma 2.
Leth0,…,hl,hl+1∈N with
(25)hl+1≥∑j=0lhj.
Then, for any given Z¯0,…,Z¯l,X¯>0, the operator ∂Z0h0⋯∂Zlhl∂X-hl+1 is a bounded linear map from G(Z¯0,…,Z¯l,X¯) into itself. Moreover,
(26)∂Z0h0⋯∂Zlhl∂X-hl+1UZ0,…,Zl,X(Z¯0,…,Z¯l,X¯)≤Z¯0-h0⋯Z¯l-hlX¯hl+1UZ0,…,Zl,X(Z¯0,…,Z¯l,X¯)
for all U(Z0,…,Zl,X)∈G(Z¯0,…,Z¯l,X¯).Proof.
LetU(Z0,…,Zl,X)∈G(Z¯0,…,Z¯l,X¯) of form (23). By definition, we can write
(27)∂Z0h0⋯∂Zlhl∂X-hl+1UZ0,…,Zl,X(Z¯0,…,Z¯l,X¯)=∑β0,…,βl,βl+1≥0∑j=0lβj+hj+βl+1-hl+1!∑j=0l+1βj!mmmmmmmmmi·∑j=0lβj+hj+βl+1-hl+1!∑j=0l+1βj!Z¯0-h0⋯Z¯l-hlX¯hl+1·Uβ0+h0,…,βl+hl,βl+1-hl+1∑j=0lβj+hj+βl+1-hl+1!·Z¯0β0+h0⋯Z¯lβl+hlX¯βl+1-hl+1.From (25), we have
(28)∑j=0lβj+hj+βl+1-hl+1!∑j=0l+1βj!≤1
for all β0,h0,…,βl,hl,βl+1,hl+1≥0. Estimate (26) follows.Lemma 3.
Leth0,…,hl,hl+1∈N. Let, for all 0≤j≤l, 0<Z¯j1<Z¯j0 and 0<X¯1<X¯0 be positive real numbers. Then, there exists C2>0 (depending on h0,…,hl+1, X¯0,X¯1, Z¯j0,Z¯j1 for 0≤j≤l) such that
(29)∂Z0h0⋯∂Zlhl∂Xhl+1UZ0,…,Zl,X(Z¯01,…,Z¯l1,X¯1)≤C2Z¯01-h0⋯Z¯l1-hlX¯1-hl+1·UZ0,…,Zl,X(Z¯00,…,Z¯l0,X¯0)
for all U(Z0,…,Zl,X)∈G(Z¯00,…,Z¯l0,X¯0).Proof.
LetU(Z0,…,Zl,X)∈G(Z¯00,…,Z¯l0,X¯0) be of form (23). By definition, we can write
(30)∂Z0h0⋯∂Zlhl∂Xhl+1UZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1=∑β0,…,βl,βl+1≥0uβ0+h0,…,βl+hl,βl+1+hl+1∑j=0l+1βj+hj!hhhhhhhhh·Z¯01-h0⋯Z¯l1-hlX¯1-hl+1hhhhhhhhh·∑j=0l+1βj+hj!∑j=0l+1βj!×Z¯01β0+h0⋯Z¯l1βl+hlhhhhhhhhhhh·X¯1βl+1+hl+1∑j=0l+1βj+hj!∑j=0l+1βj!.
Now, we take some real number 0<δ<1 such that Z¯j1<δZ¯j0 for all 0≤j≤l and X¯1<δX¯0. We get the estimates
(31)∑j=0l+1βj+hj!∑j=0l+1βj!Z¯01β0+h0⋯Z¯l1βl+hlX¯1βl+1+hl+1≤∑j=0l+1βj+∑j=0l+1hj(∑j=0l+1hj)δ(∑j=0l+1βj+∑j=0l+1hj)i·Z¯00β0+h0⋯Z¯l0βl+hlX¯0βl+1+hl+1
for all β0,h0,…,βl,hl,βl+1,hl+1≥0. Since 0<δ<1, we know that, for any real number a>0, there exists Kδ,a>0 depending on a,δ such that
(32)sup⁡n≥0n+aaδn+a≤Kδ,a.
Hence, we get a constant C2>0 depending on hj, 0≤j≤l+1 and δ with
(33)∑j=0l+1βj+∑j=0l+1hj(∑j=0l+1hj)δ(∑j=0l+1βj+∑j=0l+1hj)·Z¯00β0+h0⋯Z¯l0βl+hlX¯0βl+1+hl+1≤C2Z¯00β0+h0⋯Z¯l0βl+hlX¯0βl+1+hl+1
for all β0,h0,…,βl,hl,βl+1,hl+1≥0. As a result, gathering (30), (31), and (33), we get the lemma.Lemma 4.
LetU1(Z0,…,Zl,X),U2(Z0,…,Zl,X)∈G(Z¯0,…,Z¯l,X¯). Then, the product U1U2 belongs to G(Z¯0,…,Z¯l,X¯). Moreover, we have
(34)U1Z0,…,Zl,XU2Z0,…,Zl,X(Z¯0,…,Z¯l,X¯)≤U1Z0,…,Zl,X(Z¯0,…,Z¯l,X¯)·U2Z0,…,Zl,X(Z¯0,…,Z¯l,X¯).
In other words, the space G(Z¯0,…,Z¯l,X¯) is a Banach algebra.Proof.
Let(35)UjZ0,…,Zl,X=∑β0,…,βl,βl+1≥0uβjZ0β0⋯ZlβlXβl+1β0!⋯βl!βl+1!
belonging to G(Z¯0,…,Z¯l,X¯) for j=1,2. By definition, we can write
(36)U1Z0,…,Zl,XU2Z0,…,Zl,XZ¯0,…,Z¯l,X¯≤∑β0,…,βl,βl+1≥0∑βj1+βj2=βj,0≤j≤l+1Πj=0l+1βj!Πj=0l+1βj1Πj=0l+1βj2!1∑j=0l+1βj!mmmmmmmmm·∑βj1+βj2=βj,0≤j≤l+1Πj=0l+1βj!Πj=0l+1βj1Πj=0l+1βj2!1∑j=0l+1βj!uβ01,…,βl1,βl+111uβ02,…,βl2,βl+122·Z¯0β0⋯Z¯lβlX¯βl+1.
Besides, using the identity (1+x)∑j=0l+1‍βj=Πj=0l+1(1+x)βj and the binomial formula, we get that
(37)Πj=0l+1βj!Πj=0l+1βj1!Πj=0l+1βj2!≤∑j=0l+1βj!∑j=0l+1βj1!∑j=0l+1βj2!
for all integers βj,βj1,βj2≥0 such that βj=βj1+βj2, for 0≤j≤l+1. Therefore, inequality (34) follows from (36) and (37).In the next proposition, we state a version of the Cauchy-Kowalevski theorem.Proposition 5.
LetD1(resp., D2) be a finite subset of Nl+2(resp., Nl+1) and let S≥1 be an integer such that, for all (k0,…,kl+1)∈D1 and all (p0,…,pl)∈D2, we have
(38)S>kl+1,S≥∑j=0l+1kj,S≥∑j=0lpj.
Let D3 be a finite subset of N∖{0,1}. Let M0,X¯0>0 be given real numbers and let Z¯j0>M0, 0≤j≤l be real numbers. Let
(39)dk0,…,kl+1Z0,…,Zl,X∈GZ¯00,…,Z¯l0,X¯0,fp0,…,plZ0,…,Zl∈GZ¯00,…,Z¯l0,X¯0,emZ0,…,Zl,X∈GZ¯00,…,Z¯l0,X¯0
for all (k0,…,kl+1)∈D1, all (p0,…,pl)∈D2, and all m∈D3. For all 0≤j≤S-1, we also choose φj(Z0,…,Zl)∈G(Z¯00,…,Z¯l0,X¯0).
We consider the following Cauchy problem:(40)∂XSUZ0,…,Zl,X=∑k_=(k0,…,kl,kl+1)∈D1dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1·UZ0,…,Zl,X+∑p_=(p0,…,pl)∈D2fp_Z0,…,Zl∂Z0p0⋯∂Zlpl·∑m∈D3emZ0,…,Zl,XUZ0,…,Zl,Xm
for given initial data
(41)∂XjUZ0,…,Zl,0=φjZ0,…,Zl,0≤j≤S-1.
Then, for given real numbers Z¯j1>0, 0≤j≤l, with M0<Z¯j1<Z¯j0, one can choose 0<X¯1<X¯0 and δ>0  (which depend on Z¯j1 for 0≤j≤l, on dk_Z0,…,Zl,X(Z¯00,…,Z¯l0,X¯0) for k_∈D1, on fp_(Z0,…,Zl)(Z¯00,…,Z¯l0,X¯0) for p_∈D2, and on em(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for m∈D3) such that if
(42)φjZ0,…,ZlZ¯00,…,Z¯l0,X¯0<δ,0≤j≤S-1,
then the problem (40), (41) has a unique solution U(Z0,…,Zl,X)∈G(Z¯01,…,Z¯l1,X¯1). Moreover, there exists a constant C3>0, depending on X¯0,X¯1,Z¯j0,Z¯j1 for 0≤j≤l, on dk_(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for k_∈D1, on fp_(Z0,…,Zl)(Z¯00,…,Z¯l0,X¯0) for p_∈D2, and on em(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for m∈D3, such that
(43)UZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤δC3.Proof. We put
(44)wZ0,…,Zl,X=∑j=0S-1φjZ0,…,ZlXjj!
and we consider the map A:C[[Z0,…,Zl,X]]→C[[Z0,…,Zl,X]] defined as
(45)AHZ0,…,Zl,X=∑k_=(k0,…,kl,kl+1)∈D1dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1-S·HZ0,…,Zl,X+∑k_=(k0,…,kl,kl+1)∈D1dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1·wZ0,…,Zl,X+∑p_=(p0,…,pl)∈D2fp_Z0,…,Zl∂Z0p0⋯∂Zlpl·∑m∈D3emZ0,…,Zl,Xmii·∑m∈D3emZ0,…,Zl,X∂X-SHZ0,…,Zl,X+wZ0,…,Zl,Xm.Lemma 6.
LetZ¯j1>0, 0≤j≤l, be real numbers such that M0<Z¯j1<Zj0. Then, there exist δ>0, a real number 0<X¯1<X¯0, and a constant K4>0  (depending on Z¯j1 for 0≤j≤l, on dk_(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for k_∈D1, on fp_(Z0,…,Zl)(Z¯00,…,Z¯l0,X¯0) for p_∈D2, and on em(Z0,…,Zl,X)(Z¯00,…,Z¯l0,X¯0) for m∈D3), such that if one puts R=K4δ,(i)
we have(46)ABR⊂BR,
where BR is the closed ball of radius R, centered at 0 in G(Z¯01,…,Z¯l1,X¯1);(ii)
for allH1,H2∈BR,
(47)AH1-AH2Z¯01,…,Z¯l1,X¯1≤12H1-H2Z¯01,…,Z¯l1,X¯1.Proof.
We first show (i). We fixZ¯j1>0, 0≤j≤l, be real numbers such that M0<Z¯j1<Zj0. We also consider δ>0 for which (42) holds. Let R>0 be of the form R=K4δ for some constant K4>0. We take H∈BR where BR is the closed ball of radius R, centered at 0 in G(Z¯01,…,Z¯l1,X¯1) for some real number 0<X¯1<X¯0. From Lemmas 2 and 4, we get
(48)dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1-Sm·∂Xkl+1-SHZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1·Z¯01-k0⋯Z¯l1-klX¯1S-kl+1·HZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯00,…,Z¯l0,X¯0·Z¯01-k0⋯Z¯l1-kl×X¯1S-kl+1R.
Now, from Lemmas 3 and 4, we get a constant C4>0 (depending on k0,…,kl, X¯0,X¯1,Z¯j0,Z¯j1, for 0≤j≤l) with
(49)dk_Z0,…,Zl,X∂Z0k0⋯∂ZlklφjZ0,…,ZlXqq!Z¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1Xqq!Z¯01,…,Z¯l1,X¯1·C4Z¯01-k0⋯Z¯l1-kl×φjZ0,…,ZlZ¯00,…,Z¯l0,X¯0≤dk_Z0,…,Zl,XZ¯00,…,Z¯l0,X¯0Xqq!Z¯00,…,Z¯l0,X¯0·C4Z¯01-k0⋯Z¯l1-klδ
for all 0≤j, q≤S-1.
Using Leibniz formula, we deduce from Lemma4 that
(50)fp_Z0,…,Zp∂Z0p0⋯∂Zlplj·emZ0,…,Zl,X∂X-SHZ0,…,Zl,X+wZ0,…,Zl,Xm1m·∂X-SHZ0,…,Zl,X+wZ0,…,Zl,XmZ¯01,…,Z¯l1,X¯1≤∑∑j=0mph,j=ph,0≤h≤lp0!⋯pl!Π0≤j≤m,0≤h≤lph,j!mmmmmmmmm·fp_Z0,…,ZpZ¯01,…,Z¯l1,X¯1·∂Z0p0,0⋯∂Zlpl,0emZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1·∏j=1m∂Z0p0,j⋯∂Zlpl,j∂X-SHZ0,…,Zl,Xmmmmnnnnnnnnnnni+wZ0,…,Zl,X∂X-SZ¯01,…,Z¯l1,X¯1.
By Lemma 3, one also gets a constant C4.1>0 (depending on p0,0,…,pl,0,X¯0,X¯1,Z¯j0,Z¯j1 for 0≤j≤l) with
(51)∂Z0p0,0⋯∂Zlpl,0emZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤C4.1Z¯01-p0,0⋯Z¯l1-pl,0emZ0,…,Zl,XZ¯00,…,Z¯l0,X¯0.
By Lemma 2, one finds
(52)∂Z0p0,j⋯∂Zlpl,j∂X-SHZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤Z¯01-p0,j⋯Z¯l1-pl,jX¯1SHZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤Z¯01-p0,j⋯Z¯l1-pl,jX¯1SR.
Due to Lemma 3, we get a constant C4.2>0 (depending on p0,j,…,pl,j,X¯0,X¯1,Z¯m0,Z¯m1 for 0≤m≤l) such that
(53)∂Z0p0,j⋯∂Zlpl,jwZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤∑q=0S-1∂Z0p0,j⋯∂Zlpl,jφqZ0,…,ZlZ¯01,…,Z¯l1,X¯1Xqq!Z¯01,…,Z¯l1,X¯1≤C4.2∑q=0S-1Z¯01-p0,j⋯Z¯l1-pl,jφqZ0,…,ZlZ¯00,…,Z¯l0,X¯0·Xqq!Z¯00,…,Z¯l0,X¯0≤C4.2∑q=0S-1Z¯01-p0,j⋯Z¯l1-pl,jδXqq!Z¯00,…,Z¯l0,X¯0.
Now, we can choose0<X¯1<X¯0, δ>0 and the constant K4>0 (recall that R=K4δ) in such a way that
(54)∑k_=(k0,…,kl+1)∈D1dk_Z0,…,Zl,X(Z¯00,…,Z¯l0,X¯0)Z¯l1-kl·Z¯01-k0⋯Z¯l1-klZ¯l1-kl·X¯1S-kl+1K4δ+∑q=0S-1Xqq!(Z¯00,…,Z¯l0,X¯0)C4δ+∑p_=(p0,…,pl)∈D2∑m∈D3∑∑j=0mph,j=ph,0≤h≤lp0!⋯pl!Π0≤j≤m,0≤h≤lph,j!·fp_Z0,…,Zl(Z¯00,…,Z¯l0,X¯0)×C4.1Z¯01-p0,0⋯Z¯l1-pl,0·emZ0,…,Zl,X(Z¯00,…,Z¯l0,X¯0)·∏j=1mZ¯01-p0,j⋯Z¯l1-pl,jX¯1SK4δ∑q=0S-1Z¯01-p0,j⋯Z¯l1-pl,jδXqq!Z¯00,…,Z¯l0,X¯0mmmi+C4.2∑q=0S-1Z¯01-p0,j⋯Z¯l1-pl,jδXqq!Z¯00,…,Z¯l0,X¯0≤K4δ.
Hence, gathering (48), (49), (50), (51), (52), (53), and (54) yields inclusion (46).
We turn to the proof of (ii). As above, we fixZ¯j1>0, 0≤j≤l, be real numbers such that M0<Z¯j1<Zj0. We also consider δ>0 for which (42) holds. Let R>0 be of the form R=K4δ for some constant K4>0. We take H1,H2∈BR where BR is the closed ball of radius R, centered at 0 in G(Z¯01,…,Z¯l1,X¯1) for some real number 0<X¯1<X¯0. From Lemmas 2 and 4, we get that
(55)dk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1-Sj·H2Z0,…,Zl,X-H1Z0,…,Zl,Xdk_Z0,…,Zl,X∂Z0k0⋯∂Zlkl∂Xkl+1-SZ¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1·Z¯01-k0⋯Z¯l1-klX¯1S-kl+1·H2Z0,…,Zl,X-H1Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤dk_Z0,…,Zl,XZ¯00,…,Z¯l0,X¯0·Z¯01-k0⋯Z¯l1-klX¯1S-kl+1·H2Z0,…,Zl,X-H1Z0,…,Zl,XZ¯01,…,Z¯l1,X¯1.
Using the identity bm-am=(b-a)∑s=0m-1‍asbm-1-s for any complex numbers a,b and any integer m≥2, we can write
(56)∂X-SH1Z0,…,Zl,X+wZ0,…,Zl,Xmji-∂X-SH2Z0,…,Zl,X+wZ0,…,Zl,Xm=∂X-SH1-∂X-SH2=·∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-s.
Using Leibniz formula, we deduce from Lemma 4 that
(57)fp_Z0,…,Zl∂Z0p0⋯∂Zlplem∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-sm·Z0,…,Zl,X∂X-SH1-∂X-SH2m·∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-sZ¯01,…,Z¯l1,X¯1≤∑∑j=02ph,j=ph,0≤h≤lp0!⋯pl!Π0≤j≤2,0≤h≤lph,j!fp_Z0,…,ZlZ¯01,…,Z¯l1,X¯1·∂Z0p0,0⋯∂Zlpl,0emZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1·∂Z0p0,1⋯∂Zlpl,1∂X-SH1-∂X-SH2Z¯01,…,Z¯l1,X¯1·∂Z0p0,2⋯∂Zlpl,2∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-smii·∑s=0m-1∂X-SH2+ws∂X-SH1+wm-1-sZ¯01,…,Z¯l1,X¯1.
By Lemma 3, one also gets a constant C4.3>0 (depending on p0,0,…,pl,0,X¯0,X¯1,Z¯j0,Z¯j1 for 0≤j≤l) with
(58)∂Z0p0,0⋯∂Zlpl,0emZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤C4.3Z¯01-p0,0⋯Z¯l1-pl,0emZ0,…,Zl,XZ¯00,…,Z¯l0,X¯0.
By Lemma 2, one finds
(59)∂Z0p0,1⋯∂Zlpl,1∂X-SH1-∂X-SH2Z¯01,…,Z¯l1,X¯1≤Z¯01-p0,1⋯Z¯l1-pl,1X¯1SH1-H2Z¯01,…,Z¯l1,X¯1.
Using again Leibniz formula and Lemma 4, we can write
(60)∂Z0p0,2⋯∂Zlpl,2∂X-SH2+ws∂X-SH1+wm-1-sZ¯01,…,Z¯l1,X¯1≤∑∑j=1m-1ph,2,j=ph,2,0≤h≤lp0,2!⋯pl,2!Π1≤j≤m-1,0≤h≤lph,2,j!·∏j=1s∂Z0p0,2,j⋯∂Zlpl,0,j∂X-SH2+wZ¯01,…,Z¯l1,X¯1·∏j=s+1m-1∂Z0p0,2,j⋯∂Zlpl,0,j∂X-SH1+wZ¯01,…,Z¯l1,X¯1.
By Lemmas 2 and 3, one finds a constant C4.4>0 (depending on p0,2,j,…,pl,2,j,X¯0,X¯1,Z¯j0,Z¯j1 for 0≤j≤l) such that
(61)∂Z0p0,2,j⋯∂Zlpl,2,j∂X-SHr+wZ¯01,…,Z¯l1,X¯1≤Z¯01-p0,2,j⋯Z¯l1-pl,2,jX¯1SR+C4.4∑q=0S-1Z¯01-p0,2,j⋯Z¯l1-pl,2,jδXqq!Z¯00,…,Z¯l0,X¯0
for r=1,2.
In the following, we choose0<X¯1<X¯0 in order that
(62)∑k_=(k0,…,kl+1)∈D1dk_Z0,…,Zl,XZ¯00,…,Z¯l0,X¯0mmmmmmm·Z¯01-k0⋯Z¯l1-klX¯1S-kl+1+∑p_=(p0,…,pl)∈D2∑m∈D3∑∑j=02ph,j=ph,0≤h≤lp0!⋯pl!Π0≤j≤2,0≤h≤lph,j!mmmmmmmmmmmmimmm·fp_Z0,…,ZlZ¯00,…,Z¯l0,X¯0·C4.3Z¯01-p0,0⋯Z¯l1-pl,0emZ0,…,Zl,XZ¯00,…,Z¯l0,X¯0·Z¯01-p0,1⋯Z¯l1-pl,1X¯1S·∑s=0m-1∑∑j=1m-1ph,2,j=ph,2,0≤h≤lp0,2!⋯pl,2!Π1≤j≤m-1,0≤h≤lph,2,j!mm·∏j=1sZ¯01-p0,2,j⋯Z¯l1-pl,2,jX¯1SK4δδXqq!Z¯00,…,Z¯l0,X¯0mmmmmmi+C4.4∑q=0S-1Z¯01-p0,2,j⋯Z¯l1-pl,2,jmmmmmmmmmmmm·δXqq!Z¯00,…,Z¯l0,X¯0mm·∏j=s+1m-1Z¯01-p0,2,j⋯Z¯l1-pl,2,jX¯1SK4δδXqq!Z¯00,…,Z¯l0,X¯0mmmmmmmi+C4.4∑q=0S-1Z¯01-p0,2,j⋯Z¯l1-pl,2,jmmmmmmmmmmmmm·δXqq!Z¯00,…,Z¯l0,X¯0∑s=0m-1∑∑j=1m-1ph,2,j=ph,2,0≤h≤lp0,2!⋯pl,2!Π1≤j≤m-1,0≤h≤lph,2,j!≤12.
Taking into account all inequalities (55), (56), (57), (58), (59), (60), and (61) under constraint (62), we deduce (47).
Finally, for fixed real numbersZ¯j1>0, 0≤j≤l, such that M0<Z¯j1<Zj0, we choose 0<X¯1<X¯0, δ>0 and the constant K4>0 (recall that R=K4δ) in such a way that both constraints (54) and (62) hold. For these constants, the map A satisfies both (46) and (47).We are in a position to give the proof of Proposition5. Let Z¯j1>0, 0≤j≤l, be real numbers such that M0<Z¯j1<Zj0; we choose 0<X¯1<X¯0, δ>0 and the constant K4>0 as in Lemma 6. We have put R=δK4.From the fact thatG(Z¯01,…,Z¯l1,X¯1) equipped with the norm ·(Z¯01,…,Z¯l1,X¯1) is a Banach space, the closed ball (BR,d) for the metric d(x,y)=y-x(Z¯01,…,Z¯l1,X¯1) is a complete metric space. From Lemma 6, the map A is a contraction from (BR,d) into itself. From the classical fixed point theorem, we deduce that there exists a unique Hf(Z0,…,Zl,X)∈BR such that A(Hf)=Hf.By construction and taking into account Lemma2, the formal series
(63)UfZ0,…,Zl,X=∂X-SHfZ0,…,Zl,X+wZ0,…,Zl,X
is a solution of the problem (40), (41). Moreover, Uf(Z0,…,Zl,X)∈G(Z¯01,…,Z¯l1,X¯1) and
(64)UfZ0,…,Zl,XZ¯01,…,Z¯l1,X¯1≤δX¯1SK4+∑j=0S-1Xjj!Z¯00,…,Z¯l0,X¯0
which yields (43). Thus proof of Proposition 5 is complete.


## 2.2. Weighted Banach Spaces of Holomorphic Functions on Sectors

We denote byD(0,r) the open disc centered at 0 with radius r>0 in C. Let ϵ0>0 be a real number and let Sd be an unbounded sector in direction d∈R centered at 0 in C. By convention, these sectors do not contain the origin in C. For any open set D⊂C, we denote by O(D) the vector space of holomorphic functions on D. Let l≥1 be an integer. For all tuples β_=(β0,…,βl+1)∈Nl+2, let ρβ_≥0 be positive real numbers such that ρβ_′≤ρβ_ if |β_′|≥|β_| (where by definition |β_|=∑j=0l+1‍βj). We define Ωβ_=D(0,ρβ_)∪Sd.Definition 7.
Letb>1 be a real number and let rb(β_)=∑n=0|β_|‍1/(n+1)b for all tuples β_∈Nl+2. Let ϵ∈D(0,ϵ0)∖{0} and r,σ>0 be real numbers. We denote by Eβ_,ϵ,σ,r,Ωβ_ the vector space of all functions v∈O(Ωβ_) such that
(65)vτβ_,ϵ,σ,r,Ωβ_≔sup⁡τ∈Ωβ_vτ1+τ2ϵ2rexp⁡-σϵrrbβ_τ
is finite.Remark 8.
These norms are appropriate modifications of the norms defined in [4, 17].
In the next proposition, we study some parameter depending linear operators acting on the spacesEβ_,ϵ,σ,r,Ωβ_.Proposition 9.
Lets1,k0≥0 be integers. Let β_=(β0,…,βl+1)∈Nl+2, β~_=(β~0,…,β~l+1)∈Nl+2. Under the assumption that |β~_|>|β_|, the operator τs1∂τ-k0 defines a bounded linear map from Eβ_,ϵ,σ,r,Ωβ_ into Eβ~_,ϵ,σ,r,Ωβ~_, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, one has
(66)τs1∂τ-k0Vβ_τ,ϵβ~_,ϵ,σ,r,Ωβ~_≤Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_·ϵrs1+k0β~_+1bs1+k0s1+k0e-1σβ~_-β_s1+k0mmmmmmm+β~_+1bs1+k0+2mmmmmmm·s1+k0+2e-1σβ~_-β_s1+k0+2
for all Vβ_(τ,ϵ)∈Eβ_,ϵ,σ,r,Ωβ_.Proof.
The proof follows the same lines of arguments as Lemma 1 from [17]. By construction of the operator ∂τ-k0, one can write
(67)τs1∂τ-k0Vβ_τ,ϵ=τs1+k0∫01⋯∫01Vβ_hk0⋯h1τ,ϵ=τs1+k0n∫01⋯u·1+hk0⋯h1τ2ϵ2r=τs1+k0n∫01⋯u·exp⁡-σϵrrbβ_hk0⋯h1τ=τs1+k0n∫01⋯u·exp⁡σ/ϵrrbβ_hk0⋯h1τ1+hk0⋯h1τ2/ϵ2r=τs1+k0n∫01⋯u·Mk0h1,…,hk0-1dhk0⋯dh1,
where Mk0(h1,…,hk0-1) is a monic monomial for k0≥2, while M1=1, for all τ∈Ωβ_. We deduce that
(68)τs1∂τ-k0Vβ_τ,ϵ1+τ2ϵ2rexp⁡-σϵrrbβ~_τ≤τs1+k0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_1+τ2ϵ2r·exp⁡-σϵrrbβ~_-rbβ_τ
for all τ∈Ωβ~_⊂Ωβ_. By definition of rb, one has
(69)rbβ~_-rbβ_=∑n=|β_|+1|β~_|1n+1b≥β~_-β_β~_+1b.
Now, we recall the classical estimates: for all integers m1,m2≥0,
(70)sup⁡x≥0xm1e-m2x=m1m2m1e-m1
holds. Using (70), we deduce that
(71)τs1+k0+τs1+k0+2ϵ2rexp⁡-σϵrβ~_-β_β~_+1bτ≤ϵrs1+k0β~_+1bs1+k0s1+k0e-1σβ~_-β_s1+k0mmmmmmmmm+β~_+1bs1+k0+2mmmmmmmmm·s1+k0+2e-1σβ~_-β_s1+k0+2.
Taking into account (68), (69), and (71), we get (66).In the next proposition, we study the convolution product of functions in the spacesEβ_,ϵ,σ,r,Ωβ_.Proposition 10.
Letβ_1=(β01,…,βl+11)∈Nl+2 and β_2=(β02,…,βl+12)∈Nl+2. Let β_∈Nl+2 such that |β_|≥|β_1|+|β_2|. Then, for any Vβ_1(τ,ϵ)∈Eβ_1,ϵ,σ,r,Ωβ_1, Vβ_2(τ,ϵ)∈Eβ_2,ϵ,σ,r,Ωβ_2, the convolution product Vβ_1(τ,ϵ)*Vβ_2(τ,ϵ) belongs to Eβ_,ϵ,σ,r,Ωβ_, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, there exists a universal constant C5>0 such that
(72)∫0τVβ_1τ-s,ϵVβ_2s,ϵdsβ_,ϵ,σ,r,Ωβ_≤C5ϵrVβ_1τ,ϵβ_1,ϵ,σ,r,Ωβ_1Vβ_2τ,ϵβ_2,ϵ,σ,r,Ωβ_2
for all Vβ_1(τ,ϵ)∈Eβ_1,ϵ,σ,r,Ωβ_1, Vβ_2(τ,ϵ)∈Eβ_2,ϵ,σ,r,Ωβ_2.Proof.
We mimic the proof of Lemma 3 in [17]. One can write
(73)∫0τVβ_1τ-s,ϵVβ_2s,ϵds=∫0τVβ_1τ-s,ϵ1+τ-s2ϵ2rmmii·exp⁡-σϵrrbβ_1τ-smmii·Vβ_2s,ϵ1+s2ϵ2rexp⁡-σϵrrbβ_2smmii·exp⁡σ/ϵrrbβ_1τ-s+rbβ_2s1+τ-s2/ϵ2r1+s2/ϵ2rds
for all τ∈Ωβ_⊂Ωβ_1∩Ωβ_2. We deduce that
(74)∫0τVβ_1τ-s,ϵVβ_2s,ϵds≤Vβ_1τ,ϵβ_1,ϵ,σ,r,Ωβ_1Vβ_2τ,ϵβ_2,ϵ,σ,r,Ωβ_2·∫01τexp⁡⁡στ/ϵrrbβ_11-h+rbβ_2h1+τ2/ϵ2r1-h21+τ2/ϵ2rh2dh
for all τ∈Ωβ_. Since rb is increasing, one has
(75)rbβ_11-h+rbβ_2h≤rbβ_.
Therefore,
(76)1+τ2ϵ2rexp⁡⁡-σϵrrbβ_τ·∫01τexp⁡⁡στ/ϵrrbβ_11-h+rbβ_2h1+τ2/ϵ2r1-h21+τ2/ϵ2rh2dh≤∫011+τ2/ϵ2rτ1+τ2/ϵ2r1-h21+τ2/ϵ2rh2dh=Jτ,ϵ
for all τ∈Ωβ_. Now, from [28], we know that there exists a universal constant C5>0 such that
(77)Jϵrτ,ϵϵr≤C5
for all τ∈C, ϵ∈C*. We deduce that
(78)sup⁡τ∈CJτ,ϵϵr=sup⁡τ∈CJϵrτ,ϵϵr≤C5
for all ϵ∈C*. Finally, gathering (74), (76), and (78) yields estimates (72).Corollary 11.
Letl0≥0 be an integer. The operator ∂τ-l0 defines a bounded linear map from Eβ_,ϵ,σ,r,Ωβ_ into itself, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, there exists a constant C6>0 (depending on l0,σ) such that
(79)∂τ-l0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_≤C6ϵrl0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_
for all Vβ_(τ,ϵ)∈Eβ_,ϵ,σ,r,Ωβ_.Proof.
We carry out a similar proof as in Corollary 1 from [17]. We denote by χC the function equal to 1 on C. By definition, we put χC*1=χC and χC*l means the convolution product of χC, l-1 times for l≥2. By definition, we can write ∂τ-l0Vβ_(τ,ϵ)=(χC(τ))*l0*Vβ_(τ,ϵ). From Proposition 10, there exists a (universal) constant C5>0 such that
(80)∂τ-l0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_≤C5l0ϵrl0χCτ0_,ϵ,σ,r,Ω0_l0Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_,
where 0_=(0,…,0)∈Nl+2. By Definition 7 and using formula (70), we have that
(81)χCτ0_,ϵ,σ,r,Ω0_=sup⁡τ∈Ω0_1+τ2ϵ2rexp⁡-σϵrτ≤1+2e-1σ2.
From estimates (80) and (81), we get inequality (79).The next proposition involves bound estimates for multiplication operators of bounded holomorphic functions.Proposition 12.
Leta(τ,ϵ) be a holomorphic function on Ω0_×D(0,ϵ0)∖{0} such that there exists a constant M>0 with sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)∖{0}|a(τ,ϵ)|≤M. Then, the multiplication by a(τ,ϵ) is a bounded linear operator from Eβ_,ϵ,σ,r,Ωβ_ into itself, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, the inequality
(82)aτ,ϵVβ_τ,ϵβ_,ϵ,σ,r,Ωβ_≤MVβ_τ,ϵβ_,ϵ,σ,r,Ωβ_
holds for all Vβ_(τ,ϵ)∈Eβ_,ϵ,σ,r,Ωβ_.Proof.
The proof is a direct consequence of Definition7 for the norm ·β_,ϵ,σ,r,Ωβ_.


## 2.3. A Global Cauchy Problem

We keep the same notations as in the previous section. In the following, we introduce some definitions. LetA1 be a finite subset of N3 and let A2 be a finite subset of N2. Let l≥1 be an integer.For all(k0,k1,k2)∈A1, we denote by I(k0,k1,k2) a finite subset of N2. For all (k0,k1,k2)∈A1, all (s1,s2)∈I(k0,k1,k2), and all integers β0,βl+1≥0, we denote by as1,s2,k0,k1,k2,β0,βl+1(τ,ϵ) some holomorphic function on Ω0_×D(0,ϵ0) which satisfies the estimates: there exist constants ρ,ρ′>0, as1,s2,k0,k1,k2>0 with
(83)sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)as1,s2,k0,k1,k2,β0,βl+1τ,ϵ≤as1,s2,k0,k1,k2e-ρ′2β012ρβl+1β0!βl+1!
for all (k0,k1,k2)∈A1, we consider the series
(84)a(k0,k1,k2)τ,z,x,ϵ=∑(s1,s2)∈I(k0,k1,k2)∑β0≥0,βl+1≥0as1,s2,k0,k1,k2,β0,βl+1mmmmmmmmmmmmmm·τ,ϵτs1ϵ-s2eizβ0β0!xβl+1βl+1!
which define holomorphic functions on Ω0_×Hρ′×D(0,ρ)×D(0,ϵ0)∖{0}, where Hρ′ is defined as the following strip in C:
(85)Hρ′=z∈CIm⁡z<ρ′.
For all (l0,l1)∈A2, we denote by J(l0,l1) a finite subset of N. For all m1∈J(l0,l1) and all integers β0,βl+1≥0, we denote by αm1,l0,l1,β0,βl+1(τ,ϵ) some bounded holomorphic function on Ω0_×D(0,ϵ0) with the following estimates: there exists a constant am1,l0,l1>0 such that
(86)sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)αm1,l0,l1,β0,βl+1τ,ϵ≤am1,l0,l1e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0.For all(l0,l1)∈A2, we consider the series
(87)α(l0,l1)τ,z,x,ϵ=∑m1∈J(l0,l1)∑β0≥0,βl+1≥0αm1,l0,l1,β0,βl+1τ,ϵϵ-m1eizβ0β0!xβl+1βl+1!
which define holomorphic functions on Ω0_×Hρ′×D(0,ρ)×D(0,ϵ0)∖{0}.Letl≥1 be an integer and let ξ0=1 and ξ1,…,ξl be real algebraic numbers such that the family {1,ξ1,…,ξl} is Z-linearly independent (this means that each ξj is a real root of a polynomial Pj∈Z[X] and if there exist integers k0,…,kl∈Z such that k0+∑j=1l‍kjξj=0, then kj=0 for 0≤j≤l). Due to the classical primitive element theorem of Artin, we consider an algebraic number field K=Q(ξ) containing the numbers ξj, 1≤j≤l and we denote by h+1≥1 its degree (that is the dimension of the vector space Q(ξ) over Q). The following lemma is a direct consequence of Theorem 11 in [29].Lemma 13.
There exists a constantCξ1,…,ξl>0 (depending on ξ1,…,ξl) such that, for any k_=(k0,…,kl)∈Zl+1∖{0}, the inequality
(88)k0+k1ξ1+⋯+klξl≥Cξ1,…,ξlmax⁡j=0lkjh≥Cξ1,…,ξlk0+⋯+klh
holds.Example 14.
Letξ be an algebraic number. Assume that the degree of Q(ξ) is h+1. Then, the algebraic numbers {1,ξ,…,ξh} are Z-linearly independent and inequality (88) above holds for ξj=ξj, 0≤j≤h. In that case, one recovers Lemma 2.1 of [6].
LetS,r1,r2≥1 be integers. We put
(89)ρβ_=Cξ1,…,ξlr1/r22β0+1+β1+⋯+βl+1hr1/r2
for all β_=(β0,…,βl+1)∈Nl+2. We consider an unbounded sector Sd⊂C centered at 0 such that
(90)argτ≠2k+1r2π,0≤k≤r2-1,
for all τ∈Sd. As in the previous section, we put Ωβ_=D(0,ρβ_)∪Sd. For all 0≤j≤S-1, we choose a set of functions V(β0,…,βl,j)(τ,ϵ)∈E(β0,…,βl,j),ϵ,σ,r,Ω(β0,…,βl,j) for all β0,…,βl≥0 and we consider the formal series
(91)Vjτ,z,ϵ=∑β0,…,βl≥0Vβ0,…,βl,jτ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!
for all 0≤j≤S-1.
We consider the following Cauchy problem:(92)τr2+-i∂z+1r1∂xSVτ,z,x,ϵ=∑(k0,k1,k2)∈A1ak0,k1,k2τ,z,x,ϵ∂τ-k0∂zk1∂xk2Vτ,z,x,ϵ+∑(l0,l1)∈A2,l1≥2αl0,l1τ,z,x,ϵ∂τ-l0Vτ,z,x,ϵ*l1,
where V*1=V and V*l1, l1≥2, stands for the convolution product of V applied l1-1 times with respect to τ, for given initial conditions
(93)∂zjVτ,z,0,ϵ=Vjτ,z,ϵ,0≤j≤S-1.
In the sequel, we will need the next lemma.Lemma 15.
There exists a constantC7>0 (depending on r1,r2,Cξ1,…,ξl,Sd) such that
(94)1τr2+1+∑j=0lβjξjr1≤C71+∑j=0lβjhr1
for all τ∈Ω(β0,…,βl,βl+1+S), for all βj≥0, 0≤j≤l+1.Proof.
We putA=1+∑j=0l‍βjξj. The following partial fraction decomposition holds:
(95)1τr2+Ar1=∑k=0r2-1Akτ-Ar1/r2eiπ2k+1/r2,
where
(96)Ak=1r2e-iπ2k+1r2-1/r2Ar1-r1/r2
for all 0≤k≤r2-1. Now, there exists some constant C8>0 (depending on Sd) such that
(97)τ-1+∑j=0lβjξjr1/r2eiπ2k+1/r2≥C81+∑j=0lβjξjr1/r2
for all τ∈Ω(β0,…,βl,βl+1+S). Indeed, from (88), we know that
(98)1+∑j=0lβjξjr1/r2≥Cξ1,…,ξlr1/r21+∑j=0lβjhr1/r2>Cξ1,…,ξlr1/r21+∑j=0l+1βj+Shr1/r2
for all βj≥0, 0≤j≤l+1. Let τ∈D(0,ρ(β0,…,βl,βl+1+S)). From (98), we can write
(99)τ=h2eiθ1+∑j=0lβjξjr1/r2
for some 0≤h<1 and θ∈[0,2π). Therefore,
(100)τ-1+∑j=0lβjξjr1/r2eiπ2k+1/r2=1+∑j=0lβjξjr1/r2h2eiθ-eiπ2k+1/r2≥121+∑j=0lβjξjr1/r2.
Now, let τ∈Sd. For all k∈{0,…,r2-1}, we can write
(101)τ=s1+∑j=0lβjξjr1/r2eiπ2k+1/r2+sk,
where s≥0 and sk∈R. By construction of Sd, we have that eisk≠1 for all k∈{0,…,r2-1}. As a result, there exists a constant C9>0 (depending on Sd) with |seisk-1|≥C9, for all s≥0. Hence,
(102)τ-1+∑j=0lβjξjr1/r2eiπ2k+1/r2=1+∑j=0lβjξjr1/r2seiπ2k+1/r2+sk-eiπ2k+1/r2≥C91+∑j=0lβjξjr1/r2.
As a consequence, we get that (97) holds. On the other hand, from (88), we get that
(103)1Ar1≤1+∑j=0lβjhr1Cξ1,…,ξlr1
for all βj≥0, 0≤j≤l. Gathering (95), (97), and (103), we deduce that
(104)Akτ-Ar1/r2eiπ2k+1/r2≤1r2C8Ar1≤1r2C8Cξ1,…,ξlr11+∑j=0lβjhr1
for all 0≤k≤r2-1. The lemma follows.In the next proposition, we construct formal series solutions of (92) and (93).Proposition 16.
Under the assumption that(105)S>k2
for all (k0,k1,k2)∈A1, there exists a formal series
(106)Vτ,z,x,ϵ=∑β_=(β0,…,βl+1)∈Nl+2Vβ_τ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!
solution of (92) and (93), where the coefficients τ↦Vβ_(τ,ϵ) belong to the space O(Ωβ_) for all β_∈Nl+2 and satisfy the following recursion formula:
(107)τr2+1+∑j=0lβjξjr1Vβ0,…,βl,βl+1+Sτ,ϵβ0!⋯βl!βl+1!=∑(k0,k1,k2)∈A1∑(s1,s2)∈Ik0,k1,k2∑β01+β02=β0βl+11+βl+12=βl+1as1,s2,k0,k1,k2,β01,βl+11τ,ϵβ01!βl+11!+∑l0,l1∈A2,l1≥2∑m∈Jl0,l1hhhhhhhhhhhhhh×τs1ϵ-s2∂τ-k0Vβ02,β1,…,βl,βl+12+k2τ,ϵβ02!β1!⋯βl!βl+12!ik1β02+∑j=1lβjξjk1+∑(l0,l1)∈A2,l1≥2∑m∈J(l0,l1)∑β0-1+β00+⋯+β0l1-1=β0βj0+⋯+βjl1-1=βj,1≤j≤lβl+1-1+βl+10+⋯+βl+1l1-1=βl+1αm1,l0,l1,β0-1,βl+1-1τ,ϵϵ-m1β0-1!βl+1-1!hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh×∂τ-l0Vβ00,…,βl+10τ,ϵ*⋯*Vβ0l1-1,…,βl+1l1-1τ,ϵΠm=0l1-1Πj=0l+1βjm!
for all β0,…,βl+1≥0, all τ∈Ω(β0,…,βl,βl+1+S), and all ϵ∈D(0,ϵ0)∖{0}.Proof.
By hypothesis, we know in particular thatV(β0,…,βl,j)(τ,ϵ) belongs to O(Ω(β0,…,βl,j)) for any β0,…,βl≥0, all 0≤j≤S-1, and all ϵ∈D0,ϵ0∖{0}. Since Ωβ_′⊂Ωβ_ when |β_′|>|β_| and using Lemma 15, one gets that the functions τ↦Vβ_(τ,ϵ) which are defined by recursion (107) actually belong to O(Ωβ_) for any β_∈Nl+2, for all ϵ∈D(0,ϵ0)∖{0}. Direct computation by identification of the powers of eiz,…,eizξl and the powers of x shows that the formal series (106) is solution of (92) and (93), if its coefficients Vβ_(τ,ϵ) satisfy recursion (107).In the next proposition, we state norm inequalities for the sequenceVβ_.Proposition 17.
We consider the sequence of functionsVβ_(τ,ϵ) defined by recursion (107) for given initial data V(β0,…,βl,j)(τ,ϵ) defined above for all β0,…,βl≥0, 0≤j≤S-1. Then, for all β_∈Nl+2 and all ϵ∈D(0,ϵ0)∖{0}, the function τ↦Vβ_(τ,ϵ) belongs to Eβ_,ϵ,σ,r,Ωβ_. We put wβ_(ϵ)=Vβ_(τ,ϵ)β_,ϵ,σ,r,Ωβ_, for all β_∈Nl+2 and all ϵ∈D(0,ϵ0)∖{0}. Then, the sequence wβ_(ϵ) satisfies the following estimates. There exist constants C10>0 (depending on r1,r2,Cξ1,…,ξl,Sd) and C11>0 (depending on σ) such that
(108)wβ0,…,βl,βl+1+Sϵβ0!⋯βl!βl+1!≤∑(k0,k1,k2)∈A1∑(s1,s2)∈I(k0,k1,k2)C101+∑j=0lβjhr1·∑β01+β02=β0βl+11+βl+12=βl+1As1,s2,k0,k1,k2,β01,βl+11β01!βl+11!mmmmm·wβ02,β1,…,βl,βl+12+k2ϵβ02!β1!⋯βl!βl+12!ϵrs1+k0-s2·∑j=0l+1βj+Sbs1+k0s1+k0e-1σS-k2s1+k0mi+∑j=0l+1βj+Sbs1+k0+2s1+k0+2e-1σS-k2s1+k0+2·β0+∑j=1lβjξjk1+∑(l0,l1)∈A2,l1≥2∑m1∈J(l0,l1)C101+∑j=0lβjhr1·∑β0-1+β00+⋯+β0l1-1=β0βj0+⋯+βjl1-1=βj,1≤j≤lβl+1-1+βl+10+⋯+βl+1l1-1=βl+1Bm1,l0,l1,β0-1,βl+1-1β0-1!βl+1-1!mmmmmmmm·C11l1ϵrl0+l1-1-m1Πj=0l1-1wβ0j,…,βl+1jϵΠj=0l1-1β0j!⋯βl+1j!
for all βj≥0, 0≤j≤l+1, and all ϵ∈D(0,ϵ0)∖{0}, where
(109)As1,s2,k0,k1,k2,β01,βl+11=sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)∖0as1,s2,k0,k1,k2,β01,βl+11τ,ϵ,Bm1,l0,l1,β0-1,βl+1-1=sup⁡(τ,ϵ)∈Ω0_×D(0,ϵ0)∖0αm1,l0,l1,β0-1,βl+1-1τ,ϵ.Proof.
We apply the norm·(β0,…,βl,βl+1+S),ϵ,σ,r,Ω(β0,…,βl,βl+1+S) on the left and right hand side of equality (107) and use Propositions 9, 10, and 12, Lemma 15, and Corollary 11 in order to majorize the right hand side. Indeed, using Propositions 9, 10, and 12, Corollary 11, Lemma 15, and the estimates
(110)β02+∑j=1lβjξjk1≤β0+∑j=1lβjξjk1
we get a constant C7>0 (depending on r1,r2,Cξ1,…,ξl,Sd) such that
(111)1τr2+1+∑j=0lβjξjr1as1,s2,k0,k1,k2,β01,βl+11τ,ϵβ01!βl+11!m·τs1ϵ-s2×∂τ-k0Vβ02,β1,…,βl,βl+12+k2τ,ϵβ02!β1!⋯βl!βl+12!m·ik1β02+∑j=1lβjξjk1β0,…,βl,βl+1+S≤C71+∑j=0lβjhr1×As1,s2,k0,k1,k2,β01,βl+11β01!βl+11!·wβ02,β1,…,βl,βl+12+k2ϵβ02!β1!⋯βl!βl+12!ϵrs1+k0-s2·∑j=0l+1βj+Sbs1+k0s1+k0e-1σS-k2s1+k0mi+∑j=0l+1βj+Sb(s1+k0+2)s1+k0+2e-1σS-k2s1+k0+2·β0+∑j=1lβjξjk1
and using Propositions 10 and 12 and Corollary 11, we obtain a universal constant C5>0 and some constants C6>0 (depending on l0,σ) and C7>0 (depending on r1,r2,Cξ1,…,ξl,Sd) such that
(112)1τr2+1+∑j=0lβjξjr1αm1,l0,l1,β0-1,βl+1-1τ,ϵϵ-m1β0-1!βl+1-1!∂τ-l0Vβ00,…,βl+10τ,ϵ*⋯*Vβ0l1-1,…,βl+1l1-1τ,ϵΠm=0l1-1Πj=0l+1βjm!i·∂τ-l0Vβ00,…,βl+10τ,ϵ*⋯*Vβ0l1-1,…,βl+1l1-1τ,ϵΠm=0l1-1Πj=0l+1βjm!β0,…,βl,βl+1+S≤C71+∑j=0lβjhr1Bm1,l0,l1,β0-1,βl+1-1β0-1!βl+1-1!·C6C5l1-1ϵrl0+l1-1-m1Πj=0l1-1wβ0j,…,βl+1jϵΠj=0l1-1β0j!⋯βl+1j!.We define the following formal series:(113)As1,s2,k0,k1,k2Z0,X=∑β0,βl+1≥0As1,s2,k0,k1,k2,β0,βl+1Z0β0β0!Xβl+1βl+1!,Bm1,l0,l1Z0,X=∑β0,βl+1≥0Bm1,l0,l1,β0,βl+1Z0β0β0!Xβl+1βl+1!.
We consider the following Cauchy problem:
(114)∂XSUZ0,…,Zl,X,ϵ=∑(k0,k1,k2)∈A1∑(s1,s2)∈I(k0,k1,k2)C101+∑j=0lZj∂Zjhr1·s1+k0e-1σS-k2s1+k0∑j=0lZj∂Zj+X∂X+Sb(s1+k0)mii+s1+k0+2e-1σS-k2s1+k0+2mii·∑j=0lZj∂Zj+X∂X+Sb(s1+k0+2)·∑j=0lξjZj∂Zjk1·ϵ0rs1+k0-s2As1,s2,k0,k1,k2Z0,X∂Xk2UZ0,…,Zl,X,ϵ+∑(l0,l1)∈A2,l1≥2∑m1∈J(l0,l1)C101+∑j=0lZj∂Zjhr1·ϵ0r(l0+l1-1)-m1C11l1Bm1,l0,l1Z0,XUZ0,…,Zl,X,ϵl1
for given initial data
(115)∂XjUZ0,…,Zl,0,ϵ=∑β0≥0,…,βl≥0wβ0,…,βl,jϵZ0β0⋯Zlβlβ0!⋯βl!,0≤j≤S-1
for all ϵ∈D(0,ϵ0)∖{0}.Proposition 18.
Under the assumption that(116)S>k2+bs1+k0+2
for all (k0,k1,k2)∈A1 and all (s1,s2)∈I(k0,k1,k2), there exists a formal series
(117)UZ0,…,Zl,X,ϵ=∑β_=(β0,…,βl+1)∈Nl+2Uβ_ϵZ0β0⋯ZlβlXβl+1β0!⋯βl!βl+1!
solution of (114) and (115), where the coefficients Uβ_(ϵ) satisfy the following recursion:
(118)Uβ0,…,βl,βl+1+Sϵβ0!⋯βl!βl+1!=∑(k0,k1,k2)∈A1∑(s1,s2)∈Ik0,k1,k2C101+∑j=0lβjhr1·∑β01+β02=β0βl+11+βl+12=βl+1As1,s2,k0,k1,k2,β01,βl+11β01!βl+11!mmmmm·Uβ02,β1,…,βl,βl+12+k2ϵβ02!β1!⋯βl!βl+12!ϵ0rs1+k0-s2·∑j=0l+1βj+Sb(s1+k0)s1+k0e-1σS-k2s1+k0mi+∑j=0l+1βj+Sbs1+k0+2mi·s1+k0+2e-1σS-k2s1+k0+2·β0+∑j=1lβjξjk1+∑(l0,l1)∈A2,l1≥2∑m1∈Jl0,l1C101+∑j=0lβjhr1·∑β0-1+β00+⋯+β0l1-1=β0βj0+⋯+βjl1-1=βj,1≤j≤lβl+1-1+βl+10+⋯+βl+1l1-1=βl+1Bm1,l0,l1,β0-1,βl+1-1β0-1!βl+1-1!·C11l1ϵ0r(l0+l1-1)-m1Πj=0l1-1Uβ0j,…,βl+1jϵΠj=0l1-1β0j!⋯βl+1j!
for all βj≥0, 0≤j≤l+1, and all ϵ∈D(0,ϵ0)∖{0}.Proposition 19.
Under assumption (116) with the additional condition that
(119)rs1+k0≥s2,rl0+l1-1≥m1,l1≥2,
for all (k0,k1,k2)∈A1, all (s1,s2)∈I(k0,k1,k2), and all (l0,l1)∈A2, l1≥2 and m1∈Jl0,l1, the following inequalities hold:
(120)wβ_ϵ≤Uβ_ϵ
for all β_∈Nl+2, all ϵ∈D(0,ϵ0)∖{0}.Proof.
By assumption (115), we know that
(121)Uβ0,…,βl,jϵ=wβ0,…,βl,jϵ
for all 0≤j≤S-1, all (β0,…,βl)∈Nl+1, and all ϵ∈D(0,ϵ0)∖{0}. Therefore, we get our result by using induction from inequalities (108) and equalities (118).In the next proposition we give sufficient conditions for the formal series solutions of the Cauchy problem (92), (93) in order to define actual holomorphic functions with exponential bound estimates.Proposition 20.
We make the assumption that (119) holds. We also assume that
(122)S>bs1+k0+2+k2,S≥hr1+bs1+k0+2+k1+k2
for all (k0,k1,k2)∈A1 and all (s1,s2)∈I(k0,k1,k2).
We choose two real numbersρ1′, M0>0 such that
(123)M0>2l+2exp⁡ρ1′max⁡j=0lξj
and we take X¯0>0 and Z¯j0>M0, for 0≤j≤l. We assume that the formal series
(124)φjZ0,…,Zl,ϵ=∑β0,…,βl≥0wβ0,…,βl,jϵZ0β0⋯Zlβlβ0!⋯βl!,0≤j≤S-1,
belong to G(Z¯00,…,Z¯l0,X¯0) for all ϵ∈D(0,ϵ0)∖{0} and that there exists a constant Cφj such that sup⁡ϵ∈D(0,ϵ0)∖{0}φj(Z¯00,…,Z¯l0,X¯0)≤Cφj. As a consequence, the formal series (91) define holomorphic functions Vj(τ,z,ϵ) on the product Sd×Hρ1′×D(0,ϵ0)∖{0} and satisfy the following estimates: there exists a constant C12>0 (depending on Cφj,l) such that
(125)Vjτ,z,ϵ≤C121+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,ϵ)∈Sd×Hρ1′×D(0,ϵ0)∖{0}, and all 0≤j≤S-1.
Then, there existsδ>0 (depending on M0, Z¯j0, |ξj| for 0≤j≤l, X¯0,r1,r2,Cξ1,…,ξl,Sd, h,σ, b,S, As1,s2,k0,k1,k2(Z0,X) for (k0,k1,k2)∈A1 and (s1,s2)∈I(k0,k1,k2) and Bm1,l0,l1(Z0,X) for (l0,l1)∈A2, m1∈J(l0,l1)) such that if one assumes moreover that
(126)φjZ0,…,Zl,ϵZ¯00,…,Z¯l0,X¯0<δ,0≤j≤S-1,
for all ϵ∈D(0,ϵ0)∖{0}, the formal series (106), solution of (92) and (93), defines a holomorphic function V(τ,z,x,ϵ) on the product Sd×Hρ1′×D(0,ρ1)×D(0,ϵ0)∖{0} for some ρ1>0 and carries the next bound estimates: there exists a constant C13>0 (depending on the same as for δ given above) with
(127)Vτ,z,x,ϵ≤C131+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,x,ϵ)∈Sd×Hρ1′×D(0,ρ1)×D(0,ϵ0)∖{0}.Proof.
Sinceφj(Z0,…,Zl,ϵ) belongs to G(Z¯00,…,Z¯l0,X¯0), we get a constant C14>0 (depending on Cφj) such that
(128)Vβ0,…,βl,jτ,ϵβ0,…,βl,j,ϵ,σ,r,Ωβ0,…,βl,j≤C141Z¯00β0⋯1Z¯l0βl∑q=0lβq!
for all β0,…,βl≥0 and all 0≤j≤S-1. From the multinomial formula, we know that
(129)∑q=0lβq!≤l+1∑q=0lβqβ0!⋯βl!
for all β0,…,βl≥0. Therefore, from (128), we deduce that
(130)Vβ0,…,βl,jτ,ϵ≤C141+τ2ϵ2r-1·exp⁡σϵrrb∑q=0lβq+jτ·l+1M0∑q=0lβqβ0!⋯βl!
for all β0,…,βl≥0, 0≤j≤S-1, all τ∈Ω(β0,…,βl,j), and all ϵ∈D(0,ϵ0)∖{0}. From assumption (123), we deduce that the formal series (91) defines a holomorphic function Vj(τ,z,ϵ) on the product Sd×Hρ1′×D(0,ϵ0)∖{0} and satisfies
(131)Vjτ,z,ϵ≤C141+τ2ϵ2r-1exp⁡σϵrζbτ·∑β0,…,βl≥0l+1exp⁡⁡ρ1′max⁡q=0l⁡ξqM0∑q=0lβq≤2l+1C141+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,ϵ)∈Sd×Hρ1′×D(0,ϵ0)∖{0} and all 0≤j≤S-1.
Under assumptions (119) and (122) together with (126), we see that the hypotheses of Proposition 5 are fulfilled for the Cauchy problem (114), (115). Therefore, we deduce that the formal solution U(Z0,…,Zl,X,ϵ) of (114), (115) constructed in Proposition 18 belongs to the space G(Z¯01,…,Z¯l1,X¯1) for some 0<X¯1<X¯0 and for some Z¯j1>M0. Moreover, we get a constant C15>0 (depending on Z¯j0,Z¯j1,|ξj| for 0≤j≤l, X¯0,r1,r2,Cξ1,…,ξl,Sd, h,σ,b,S, As1,s2,k0,k1,k2(Z0,X) for (k0,k1,k2)∈A1 and (s1,s2)∈I(k0,k1,k2) and Bm1,l0,l1(Z0,X) for (l0,l1)∈A2, m1∈J(l0,l1)) such that
(132)UZ0,…,Zl,X,ϵZ¯01,…,Z¯l1≤δC15
for all ϵ∈D(0,ϵ0)∖{0}. In particular, we deduce that
(133)Uβ0,…,βl+1ϵ≤δC151Z¯01β0⋯1Z¯l1βl1X¯1βl+1∑j=0l+1βj!
for all β0,…,βl+1≥0. Gathering (120) and (133) yields
(134)Vβ_τ,ϵβ_,ϵ,σ,r,Ωβ_≤δC151Z¯01β0⋯1Z¯l1βl1X¯1βl+1∑j=0l+1βj!
for all β_=(β0,…,βl+1)∈Nl+2. Again by the multinomial formula, we have that
(135)∑j=0l+1βj!≤l+2∑j=0l+1βjβ0!⋯βl+1!
for all β0,…,βl+1≥0. Hence, from (134), we get that
(136)Vβ_τ,ϵ≤δC151+τ2ϵ2r-1exp⁡σϵrrbβ_τ·l+2M0∑j=0lβjl+2X¯1βl+1β0!⋯βl+1!
for all β0,…,βl+1≥0, all τ∈Ωβ_, and all ϵ∈D(0,ϵ0)∖{0}. We deduce that the formal series (106) defines a holomorphic function V(τ,z,x,ϵ) on the product Sd×Hρ1′×D(0,X¯1/2(l+2))×D(0,ϵ0)∖{0} and satisfies
(137)Vτ,z,x,ϵ≤δC151+τ2ϵ2r-1exp⁡σϵrζbτ·∑β0,…,βl+1≥0l+2exp⁡⁡ρ1′max⁡j=0lξjM0∑j=0lβj·l+2xX¯1βl+1≤2l+2δC151+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,x,ϵ)∈Sd×Hρ1′×D(0,X¯1/2(l+2))×D(0,ϵ0)∖{0}.


## 3. Analytic Solutions in a Complex Parameter of a Singular Cauchy Problem

### 3.1. Laplace Transform and Asymptotic Expansions

We recall the definition of Borel summability of formal series with coefficients in a Banach space; see [18].Definition 21.
A formal series(138)X^t=∑j=0∞ajj!tj∈Et
with coefficients in a Banach space (E,·E) is said to be 1-summable with respect to t in the direction d∈[0,2π) if(i)
there existsρ∈R+ such that the following formal series, called formal Borel transform of X^ of order 1
(139)BX^τ=∑j=0∞ajτjj!2∈Eτ,
is absolutely convergent for |τ|<ρ;(ii)
there existsδ>0 such that the series B(X^)(τ) can be analytically continued with respect to τ in a sector Sd,δ={τ∈C*:|d-arg(τ)|<δ}. Moreover, there exist C>0 and K>0 such that
(140)BX^τE≤CeKτ
for all τ∈Sd,δ.If this is so, the vector valued Laplace transform of order 1 ofB(X^)(τ) in the direction d is defined by
(141)LdBX^t=t-1∫LγBX^τe-τ/tdτ,
along a half-line Lγ=R+eiγ⊂Sd,δ∪{0}, where γ depends on t and is chosen in such a way that cos⁡(γ-arg(t))≥δ1>0, for some fixed δ1, for all t in any sector
(142)Sd,θ,R=t∈C*:t<R,d-argt<θ2,
where π<θ<π+2δ and 0<R<δ1/K. The function Ld(B(X^))(t) is called the 1-sum of the formal series X^(t) in the direction d and defines a holomorphic bounded function on the sector Sd,θ,R. Moreover, it has the formal series X^(t) as Gevrey asymptotic expansion of order 1 with respect to t on Sd,θ,R. This means that, for all θ1<θ, there exist C,M>0 such that
(143)LdBX^t-∑p=0n-1app!tpE≤CMnn!tn
for all n≥1 and all t∈Sd,θ1,R.In the next proposition, we give some well-known identities for the Borel transform that will be useful in the sequel.Proposition 22.
LetX^(t)=∑n≥0‍antn/n! and G^(t)=∑n≥0‍bntn/n! be formal series in E[[t]]. We have the following equalities as formal series in E[[τ]]:
(144)τ∂τ2+∂τBX^τ=B∂tX^tτ,∂τ-1BX^τ=BtX^tτ,τBX^τ=Bt2∂t+tX^tτ,∫0τBX^τ-sBG^sds=BtX^tG^tτ.Proof.
By a direct computation, we have the following expansions from which Proposition22 follows:
(145)∂tX^t=∑n≥0an+1tnn!,τ∂τ2+∂τBX^τ=∑n≥0an+1τnn!2,tX^t=∑n≥1nan-1tnn!,∂τ-1BX^τ=∑n≥1nan-1τnn!2,t2∂t+tX^t=∑n≥1n2an-1tnn!,τBX^τ=∑n≥1n2an-1τnn!2,tX^tG^t=∑n≥1∑l+m=n-1n!l!m!albmtnn!,∫0τBX^τ-sBG^sds=∑n≥1∑l+m=n-1n!l!m!albmτnn!2.


### 3.2. Analytic Solutions of Some Singular Cauchy Problem

LetS,r1,r2≥1 be integers. Let S be a finite subset of N4 and N a finite subset of N2. For all (s,k0,k1,k2)∈S and all integers β0,βl+1≥0, we denote by bs,k0,k1,k2,β0,βl+1(ϵ) some holomorphic function on D(0,ϵ0) which satisfies the next estimates: there exist constants ρ,ρ′>0, bs,k0,k1,k2>0 with
(146)sup⁡ϵ∈D(0,ϵ0)bs,k0,k1,k2,β0,βl+1ϵ≤bs,k0,k1,k2e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0. Likewise, for all (l0,l1)∈N and all integers β0,βl+1≥0, we denote by cl0,l1,β0,βl+1(ϵ) some holomorphic function on D(0,ϵ0) with the following estimates: there exists a constant cl0,l1>0 with
(147)sup⁡ϵ∈D(0,ϵ0)cl0,l1,β0,βl+1ϵ≤cl0,l1e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0.For all(s,k0,k1,k2)∈S and all (l0,l1)∈N, we consider the series
(148)bs,k0,k1,k2z,x,ϵ=∑β0,βl+1≥0bs,k0,k1,k2,β0,βl+1ϵeizβ0β0!xβl+1βl+1!,cl0,l1z,x,ϵ=∑β0,βl+1≥0cl0,l1,β0,βl+1ϵeizβ0β0!xβl+1βl+1!
which define bounded holomorphic functions on Hρ′×D(0,ρ)×D(0,ϵ0).We consider the following singular Cauchy problem:(149)T2∂T+Tr2+-i∂z+1r1∂xSYUd,D0,ϵ0∖0T,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵϵrk0-sTsmmmmmmmm·∂Tk0∂zk1∂xk2YUd,D0,ϵ0∖0T,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵϵ-rl0+l1-1Tl0+l1-1mmmmmmm·YUd,D0,ϵ0∖0T,z,x,ϵl1
for given initial data
(150)∂xjYUd,D0,ϵ0∖0T,z,0,ϵ=YUd,D0,ϵ0∖0,jT,z,ϵ,0≤j≤S-1.
The initial conditions are constructed as follows. Let Ud be an unbounded sector such that
(151)argτ≠2k+1r2π
for all 0≤k≤r2-1 and all τ∈Ud. For all 0≤j≤S-1 and all (β0,…,βl)∈Nl+1, let VUd,(β0,…,βl,j)(τ,ϵ) be a function such that
(152)VUd,β0,…,βl,jτ,ϵ∈Eβ0,…,βl,j,ϵ,σ,r,D0,ρβ0,…,βl,j∪Ud
for all ϵ∈D(0,ϵ0)∖{0}.We choose two real numbersρ1′,M0>0 such that
(153)M0>2l+2exp⁡ρ1′max⁡j=0lξj
and we take X¯0>0 and Z¯j0>M0, for all 0≤j≤l. We make the assumption that the formal series
(154)φd,jZ0,…,Zl,ϵ=∑β0,…,βl≥0VUd,β0,…,βl,jτ,ϵβ0,…,βl,j,ϵ,σ,r,D0,ρβ0,…,βl,j∪Udmmmmm·Z0β0⋯Zlβlβ0!⋯βl!
belongs to the Banach space G(Z¯00,…,Z¯l0,X¯0) for all ϵ∈D(0,ϵ0)∖{0} and all 0≤j≤S-1. Moreover, we assume that there exists a constant Cφd,j>0 such that sup⁡ϵ∈D(0,ϵ0)∖{0}φd,j(Z¯00,…,Z¯l0,X¯0)≤Cφd,j.Let(155)VUd,β0,…,βl,jτ,ϵ=∑m≥0χm,β0,…,βl,jϵm!2τm
be its Taylor expansion with respect to τ on D(0,ρ(β0,…,βl,j)), for all ϵ∈D(0,ϵ0)∖{0}. We consider the formal series
(156)Y^β0,…,βl,jT,ϵ=∑m≥0χm,β0,…,βl,jϵm!Tm
for all ϵ∈D(0,ϵ0)∖{0}. For all (β0,…,βl,j) we define YUd,D(0,ϵ0)∖{0},(β0,…,βl,j)(T,ϵ) as the 1-sum of Y^(β0,…,βl,j)(T,ϵ) in the direction d. From the fact that τ↦VUd,(β0,…,βl,j)(τ,ϵ) belongs to E(β0,…,βl,j),ϵ,σ,r,D(0,ρ(β0,…,βl,j))∪Ud, for all ϵ∈D(0,ϵ0)∖{0}, we get that YUd,D(0,ϵ0)∖{0},(β0,…,βl,j)(T,ϵ) defines a holomorphic function for all T∈Ud,θ,h|ϵ|r and all ϵ∈D(0,ϵ0)∖{0}, where
(157)Ud,θ,hϵr=T∈C:T<hϵr,d-argT<θ2,
for some θ>π and some constant h>0 (independent of ϵ and β0,…,βl), for all 0≤j≤S-1. The initial data are defined as the formal series
(158)YUd,D0,ϵ0∖0,jT,z,ϵ=∑β0,…,βl≥0YUd,D0,ϵ0∖0,β0,…,βl,jT,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!,mmmmmmmmmmmmmmmiimmmmmm0≤j≤S-1,
which actually define holomorphic functions on the domain Ud,θ,h|ϵ|r×Hρ1′×D(0,ϵ0)∖{0}. Indeed from hypotheses (153) and (154) and the multinomial formula (129), following the first part of the proof of Proposition 20, we get that there exists a constant C16>0 such that
(159)∑β0,…,βl≥0VUd,β0,…,βl,jτ,ϵexp⁡⁡iz∑j=0lβjξjβ0!⋯βl!≤C161+τ2ϵ2r-1exp⁡σϵrζbτ·∑β0,…,βl≥0l+1exp⁡⁡ρ1′max⁡j=0lξjM0∑j=0lβj≤2l+1C161+τ2ϵ2r-1exp⁡σϵrζbτ
for all τ∈Ud, z∈Hρ1′ and ϵ∈D(0,ϵ0)∖{0}.We get the following result.Proposition 23.
Let the initial data be constructed as above. We make the following assumptions. For all(s,k0,k1,k2)∈S and all (l0,l1)∈N, we have that
(160)s≥2k0,S>k2,S>bs-k0+2+k2,S≥hr1+bs-k0+2+k1+k2,l1≥2.
Then, there exists a constant I>0 (independent of ϵ) such that if one assumes that
(161)φd,jZ0,…,Zl,ϵZ¯00,…,Z¯l0,X¯0<I,
where φd,j is defined in (154), for all ϵ∈D(0,ϵ0)∖{0}, the problem (149), (150) has a solution
(162)YUd,D0,ϵ0∖0T,z,x,ϵ=∑β0,…,βl,βl+1≥0YUd,D0,ϵ0∖0,β0,…,βl,βl+1T,ϵnnnnnnnnnn·exp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!
which defines a bounded holomorphic function on Ud,θ,h′|ϵ|r×Hρ1′×D(0,ρ1) for some h′>0 and ρ1>0, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, each function YUd,D(0,ϵ0)∖{0},(β0,…,βl,βl+1)(T,ϵ) can be written as a Laplace transform of order 1 in the direction d of a function τ↦VUd,(β0,…,βl,βl+1)(τ,ϵ) which is holomorphic on D(0,ρ(β0,…,βl,βl+1))∪Ud and satisfies the estimates: there exist two constants C17>0, K17>0 (both independent of ϵ) such that
(163)VUd,β0,…,βl,βl+1τ,ϵ≤C17exp⁡σϵrζbτl+2M0∑j=0lβjK17βl+1β0!⋯βl+1!
for all τ∈D(0,ρ(β0,…,βl,βl+1))∪Ud and ϵ∈D(0,ϵ0)∖{0}.Proof. One considers a formal series
(164)Y^T,z,x,ϵ=∑β0,…,βl,βl+1≥0Y^β0,…,βl,βl+1T,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!
solution of (149), with initial data
(165)∂xjY^T,z,0,ϵ=Y^jT,z,ϵ=∑β0,…,βl≥0Y^β0,…,βl,jT,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!,mmmmmmmmmmmmmmmiii0≤j≤S-1,
for all ϵ∈D(0,ϵ0)∖{0} where Y^(β0,…,βl,j)(T,ϵ) are defined in (156). We consider the formal Borel transform of Y^(T,z,x,ϵ) of order 1 with respect to T denoted by
(166)V^τ,z,x,ϵ=∑β0,…,βl,βl+1≥0V^β0,…,βl,βl+1τ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!,
where, by construction, V^(β0,…,βl,βl+1)(τ,ϵ) is the Borel transform of order 1 with respect to T of the formal series Y^(β0,…,βl,βl+1)(T,ϵ), for all ϵ∈D(0,ϵ0)∖{0}.From the identities of Proposition22, we get that V^(τ,z,x,ϵ) satisfies the following singular Cauchy problem:
(167)τr2+-i∂z+1r1∂xSV^τ,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵϵrk0-s∂τ-smmmmmmm·τ∂τ2+∂τk0∂zk1∂xk2V^τ,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵϵ-rl0+l1-1∂τ-l0V^τ,z,x,ϵ*l1
with initial data
(168)∂xjV^τ,z,0,ϵ=∑β0,…,βl≥0V(β0,…,βl,j)τ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!,mmmmmmmmmmmmmmmmiim0≤j≤S-1,
where V(β0,…,βl,j)(τ,ϵ) are defined in (152), for all ϵ∈D(0,ϵ0)∖{0}. In the following, we rewrite (167) using the two following technical lemmas. Their proofs can be found in [17, Lemmas 5 and 6]. Therefore we omit them.Lemma 24.
For allk0≥1, there exist constants ak,k0∈N, k0≤k≤2k0, such that
(169)τ∂τ2+∂τk0uτ=∑k=k02k0ak,k0τk-k0∂τkuτ
for all holomorphic functions u:Ω→C on an open set Ω⊂C.Lemma 25.
Leta,b,c≥0 be positive integers such that a≥b and a≥c. We put δ=a+b-c. Then, for all holomorphic functions u:Ω→C, the function ∂τ-a(τb∂τcu(τ)) can be written in the form
(170)∂τ-aτb∂τcuτ=∑(b′,c′)∈Oδαb′,c′τb′∂τc′uτ,
where Oδ is a finite subset of Z2 such that, for all (b′,c′)∈Oδ, b′-c′=δ, b′≥0, c′≤0, and αb′,c′∈Z.Using the latter Lemmas24 and 25 together with assumption (160), we can rewrite (167) in the form
(171)τr2+-i∂z+1r1∂xSV^τ,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵϵrk0-smmmmmmmi·∑(r′,p′)∈Os-k0αr′,p′τr′∂τ-p′∂zk1∂xk2V^τ,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵϵ-rl0+l1-1∂τ-l0V^τ,z,x,ϵ*l1,
where Os-k0 is a finite subset of N2 such that, for all r′,p′∈Os-k0, we have r′+p′=s-k0 and αr′,p′∈Z, for the given initial data
(172)∂xjV^τ,z,0,ϵ=∑β0,…,βl≥0Vβ0,…,βl,jτ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!,mmmmmmmmmmmmmimimmii0≤j≤S-1.From assumption (160), we deduce that assumptions (119) and (122) of Proposition 20 are fulfilled for (171). Hence, from Proposition 20, we deduce the existence of a constant I>0 (independent of ϵ) such that if inequality (161) holds, then the formal series V^(τ,z,x,ϵ) solution of (171) and (172) defines a holomorphic function VUd(τ,z,x,ϵ) on the product Ud×Hρ1′×D(0,ρ1)×D(0,ϵ0)∖{0} which satisfies the next bound estimates: there exists a constant C18>0 such that
(173)VUdτ,z,x,ϵ≤C181+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,x,ϵ)∈Ud×Hρ1′×D(0,ρ1)×D(0,ϵ0)∖{0}. Moreover, from the proof of Proposition 20 (especially formula (136)), we also get that each formal series V^(β0,…,βl,βl+1)(τ,ϵ) defines a holomorphic function VUd,(β0,…,βl,βl+1)(τ,ϵ) on (Ud∪D(0,ρ(β0,…,βl+1)))×D(0,ϵ0)∖{0} with the following estimates: there exist two constants C19>0, K19>0 such that
(174)VUd,β0,…,βl,βl+1τ,ϵ≤IC19exp⁡σϵrrbβ0,…,βl+1τl+2M0∑j=0lβj·K19βl+1β0!⋯βl+1!
for all τ∈D(0,ρ(β0,…,βl+1))∪Ud and ϵ∈D(0,ϵ0)∖{0}. From (174), we get that each formal series Y^(β0,…,βl,βl+1)(T,ϵ) is 1-summable with respect to the fact that T is the direction d and its 1-sum denoted by YUd,D(0,ϵ0)∖{0},(β0,…,βl,βl+1)(T,ϵ) can be written as Laplace transform of order 1 in the direction d of the function τ↦VUd,(β0,…,βl,βl+1)(τ,ϵ). Moreover, again by (174), we deduce that the series
(175)YUd,D0,ϵ0∖0T,z,x,ϵ=∑β0,…,βl,βl+1≥0YUd,D0,ϵ0∖0,β0,…,βl,βl+1T,ϵmmmmmmmm·exp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!
defines a bounded holomorphic function on Ud,θ,h′|ϵ|r×Hρ1′×D(0,ρ1) for some h′>0 and ρ1>0, for all ϵ∈D(0,ϵ0)∖{0}. Finally, from the algebraic properties of the κ-summation procedure (see [18] Section 6.3) since Y^(T,z,x,ϵ) formally solves (149), we deduce that YUd,D(0,ϵ0)∖{0}(T,z,x,ϵ) is an actual solution of the Cauchy problem (149), (150).


## 3.1. Laplace Transform and Asymptotic Expansions

We recall the definition of Borel summability of formal series with coefficients in a Banach space; see [18].Definition 21.
A formal series(138)X^t=∑j=0∞ajj!tj∈Et
with coefficients in a Banach space (E,·E) is said to be 1-summable with respect to t in the direction d∈[0,2π) if(i)
there existsρ∈R+ such that the following formal series, called formal Borel transform of X^ of order 1
(139)BX^τ=∑j=0∞ajτjj!2∈Eτ,
is absolutely convergent for |τ|<ρ;(ii)
there existsδ>0 such that the series B(X^)(τ) can be analytically continued with respect to τ in a sector Sd,δ={τ∈C*:|d-arg(τ)|<δ}. Moreover, there exist C>0 and K>0 such that
(140)BX^τE≤CeKτ
for all τ∈Sd,δ.If this is so, the vector valued Laplace transform of order 1 ofB(X^)(τ) in the direction d is defined by
(141)LdBX^t=t-1∫LγBX^τe-τ/tdτ,
along a half-line Lγ=R+eiγ⊂Sd,δ∪{0}, where γ depends on t and is chosen in such a way that cos⁡(γ-arg(t))≥δ1>0, for some fixed δ1, for all t in any sector
(142)Sd,θ,R=t∈C*:t<R,d-argt<θ2,
where π<θ<π+2δ and 0<R<δ1/K. The function Ld(B(X^))(t) is called the 1-sum of the formal series X^(t) in the direction d and defines a holomorphic bounded function on the sector Sd,θ,R. Moreover, it has the formal series X^(t) as Gevrey asymptotic expansion of order 1 with respect to t on Sd,θ,R. This means that, for all θ1<θ, there exist C,M>0 such that
(143)LdBX^t-∑p=0n-1app!tpE≤CMnn!tn
for all n≥1 and all t∈Sd,θ1,R.In the next proposition, we give some well-known identities for the Borel transform that will be useful in the sequel.Proposition 22.
LetX^(t)=∑n≥0‍antn/n! and G^(t)=∑n≥0‍bntn/n! be formal series in E[[t]]. We have the following equalities as formal series in E[[τ]]:
(144)τ∂τ2+∂τBX^τ=B∂tX^tτ,∂τ-1BX^τ=BtX^tτ,τBX^τ=Bt2∂t+tX^tτ,∫0τBX^τ-sBG^sds=BtX^tG^tτ.Proof.
By a direct computation, we have the following expansions from which Proposition22 follows:
(145)∂tX^t=∑n≥0an+1tnn!,τ∂τ2+∂τBX^τ=∑n≥0an+1τnn!2,tX^t=∑n≥1nan-1tnn!,∂τ-1BX^τ=∑n≥1nan-1τnn!2,t2∂t+tX^t=∑n≥1n2an-1tnn!,τBX^τ=∑n≥1n2an-1τnn!2,tX^tG^t=∑n≥1∑l+m=n-1n!l!m!albmtnn!,∫0τBX^τ-sBG^sds=∑n≥1∑l+m=n-1n!l!m!albmτnn!2.


## 3.2. Analytic Solutions of Some Singular Cauchy Problem

LetS,r1,r2≥1 be integers. Let S be a finite subset of N4 and N a finite subset of N2. For all (s,k0,k1,k2)∈S and all integers β0,βl+1≥0, we denote by bs,k0,k1,k2,β0,βl+1(ϵ) some holomorphic function on D(0,ϵ0) which satisfies the next estimates: there exist constants ρ,ρ′>0, bs,k0,k1,k2>0 with
(146)sup⁡ϵ∈D(0,ϵ0)bs,k0,k1,k2,β0,βl+1ϵ≤bs,k0,k1,k2e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0. Likewise, for all (l0,l1)∈N and all integers β0,βl+1≥0, we denote by cl0,l1,β0,βl+1(ϵ) some holomorphic function on D(0,ϵ0) with the following estimates: there exists a constant cl0,l1>0 with
(147)sup⁡ϵ∈D(0,ϵ0)cl0,l1,β0,βl+1ϵ≤cl0,l1e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0.For all(s,k0,k1,k2)∈S and all (l0,l1)∈N, we consider the series
(148)bs,k0,k1,k2z,x,ϵ=∑β0,βl+1≥0bs,k0,k1,k2,β0,βl+1ϵeizβ0β0!xβl+1βl+1!,cl0,l1z,x,ϵ=∑β0,βl+1≥0cl0,l1,β0,βl+1ϵeizβ0β0!xβl+1βl+1!
which define bounded holomorphic functions on Hρ′×D(0,ρ)×D(0,ϵ0).We consider the following singular Cauchy problem:(149)T2∂T+Tr2+-i∂z+1r1∂xSYUd,D0,ϵ0∖0T,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵϵrk0-sTsmmmmmmmm·∂Tk0∂zk1∂xk2YUd,D0,ϵ0∖0T,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵϵ-rl0+l1-1Tl0+l1-1mmmmmmm·YUd,D0,ϵ0∖0T,z,x,ϵl1
for given initial data
(150)∂xjYUd,D0,ϵ0∖0T,z,0,ϵ=YUd,D0,ϵ0∖0,jT,z,ϵ,0≤j≤S-1.
The initial conditions are constructed as follows. Let Ud be an unbounded sector such that
(151)argτ≠2k+1r2π
for all 0≤k≤r2-1 and all τ∈Ud. For all 0≤j≤S-1 and all (β0,…,βl)∈Nl+1, let VUd,(β0,…,βl,j)(τ,ϵ) be a function such that
(152)VUd,β0,…,βl,jτ,ϵ∈Eβ0,…,βl,j,ϵ,σ,r,D0,ρβ0,…,βl,j∪Ud
for all ϵ∈D(0,ϵ0)∖{0}.We choose two real numbersρ1′,M0>0 such that
(153)M0>2l+2exp⁡ρ1′max⁡j=0lξj
and we take X¯0>0 and Z¯j0>M0, for all 0≤j≤l. We make the assumption that the formal series
(154)φd,jZ0,…,Zl,ϵ=∑β0,…,βl≥0VUd,β0,…,βl,jτ,ϵβ0,…,βl,j,ϵ,σ,r,D0,ρβ0,…,βl,j∪Udmmmmm·Z0β0⋯Zlβlβ0!⋯βl!
belongs to the Banach space G(Z¯00,…,Z¯l0,X¯0) for all ϵ∈D(0,ϵ0)∖{0} and all 0≤j≤S-1. Moreover, we assume that there exists a constant Cφd,j>0 such that sup⁡ϵ∈D(0,ϵ0)∖{0}φd,j(Z¯00,…,Z¯l0,X¯0)≤Cφd,j.Let(155)VUd,β0,…,βl,jτ,ϵ=∑m≥0χm,β0,…,βl,jϵm!2τm
be its Taylor expansion with respect to τ on D(0,ρ(β0,…,βl,j)), for all ϵ∈D(0,ϵ0)∖{0}. We consider the formal series
(156)Y^β0,…,βl,jT,ϵ=∑m≥0χm,β0,…,βl,jϵm!Tm
for all ϵ∈D(0,ϵ0)∖{0}. For all (β0,…,βl,j) we define YUd,D(0,ϵ0)∖{0},(β0,…,βl,j)(T,ϵ) as the 1-sum of Y^(β0,…,βl,j)(T,ϵ) in the direction d. From the fact that τ↦VUd,(β0,…,βl,j)(τ,ϵ) belongs to E(β0,…,βl,j),ϵ,σ,r,D(0,ρ(β0,…,βl,j))∪Ud, for all ϵ∈D(0,ϵ0)∖{0}, we get that YUd,D(0,ϵ0)∖{0},(β0,…,βl,j)(T,ϵ) defines a holomorphic function for all T∈Ud,θ,h|ϵ|r and all ϵ∈D(0,ϵ0)∖{0}, where
(157)Ud,θ,hϵr=T∈C:T<hϵr,d-argT<θ2,
for some θ>π and some constant h>0 (independent of ϵ and β0,…,βl), for all 0≤j≤S-1. The initial data are defined as the formal series
(158)YUd,D0,ϵ0∖0,jT,z,ϵ=∑β0,…,βl≥0YUd,D0,ϵ0∖0,β0,…,βl,jT,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!,mmmmmmmmmmmmmmmiimmmmmm0≤j≤S-1,
which actually define holomorphic functions on the domain Ud,θ,h|ϵ|r×Hρ1′×D(0,ϵ0)∖{0}. Indeed from hypotheses (153) and (154) and the multinomial formula (129), following the first part of the proof of Proposition 20, we get that there exists a constant C16>0 such that
(159)∑β0,…,βl≥0VUd,β0,…,βl,jτ,ϵexp⁡⁡iz∑j=0lβjξjβ0!⋯βl!≤C161+τ2ϵ2r-1exp⁡σϵrζbτ·∑β0,…,βl≥0l+1exp⁡⁡ρ1′max⁡j=0lξjM0∑j=0lβj≤2l+1C161+τ2ϵ2r-1exp⁡σϵrζbτ
for all τ∈Ud, z∈Hρ1′ and ϵ∈D(0,ϵ0)∖{0}.We get the following result.Proposition 23.
Let the initial data be constructed as above. We make the following assumptions. For all(s,k0,k1,k2)∈S and all (l0,l1)∈N, we have that
(160)s≥2k0,S>k2,S>bs-k0+2+k2,S≥hr1+bs-k0+2+k1+k2,l1≥2.
Then, there exists a constant I>0 (independent of ϵ) such that if one assumes that
(161)φd,jZ0,…,Zl,ϵZ¯00,…,Z¯l0,X¯0<I,
where φd,j is defined in (154), for all ϵ∈D(0,ϵ0)∖{0}, the problem (149), (150) has a solution
(162)YUd,D0,ϵ0∖0T,z,x,ϵ=∑β0,…,βl,βl+1≥0YUd,D0,ϵ0∖0,β0,…,βl,βl+1T,ϵnnnnnnnnnn·exp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!
which defines a bounded holomorphic function on Ud,θ,h′|ϵ|r×Hρ1′×D(0,ρ1) for some h′>0 and ρ1>0, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, each function YUd,D(0,ϵ0)∖{0},(β0,…,βl,βl+1)(T,ϵ) can be written as a Laplace transform of order 1 in the direction d of a function τ↦VUd,(β0,…,βl,βl+1)(τ,ϵ) which is holomorphic on D(0,ρ(β0,…,βl,βl+1))∪Ud and satisfies the estimates: there exist two constants C17>0, K17>0 (both independent of ϵ) such that
(163)VUd,β0,…,βl,βl+1τ,ϵ≤C17exp⁡σϵrζbτl+2M0∑j=0lβjK17βl+1β0!⋯βl+1!
for all τ∈D(0,ρ(β0,…,βl,βl+1))∪Ud and ϵ∈D(0,ϵ0)∖{0}.Proof. One considers a formal series
(164)Y^T,z,x,ϵ=∑β0,…,βl,βl+1≥0Y^β0,…,βl,βl+1T,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!
solution of (149), with initial data
(165)∂xjY^T,z,0,ϵ=Y^jT,z,ϵ=∑β0,…,βl≥0Y^β0,…,βl,jT,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!,mmmmmmmmmmmmmmmiii0≤j≤S-1,
for all ϵ∈D(0,ϵ0)∖{0} where Y^(β0,…,βl,j)(T,ϵ) are defined in (156). We consider the formal Borel transform of Y^(T,z,x,ϵ) of order 1 with respect to T denoted by
(166)V^τ,z,x,ϵ=∑β0,…,βl,βl+1≥0V^β0,…,βl,βl+1τ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!,
where, by construction, V^(β0,…,βl,βl+1)(τ,ϵ) is the Borel transform of order 1 with respect to T of the formal series Y^(β0,…,βl,βl+1)(T,ϵ), for all ϵ∈D(0,ϵ0)∖{0}.From the identities of Proposition22, we get that V^(τ,z,x,ϵ) satisfies the following singular Cauchy problem:
(167)τr2+-i∂z+1r1∂xSV^τ,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵϵrk0-s∂τ-smmmmmmm·τ∂τ2+∂τk0∂zk1∂xk2V^τ,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵϵ-rl0+l1-1∂τ-l0V^τ,z,x,ϵ*l1
with initial data
(168)∂xjV^τ,z,0,ϵ=∑β0,…,βl≥0V(β0,…,βl,j)τ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!,mmmmmmmmmmmmmmmmiim0≤j≤S-1,
where V(β0,…,βl,j)(τ,ϵ) are defined in (152), for all ϵ∈D(0,ϵ0)∖{0}. In the following, we rewrite (167) using the two following technical lemmas. Their proofs can be found in [17, Lemmas 5 and 6]. Therefore we omit them.Lemma 24.
For allk0≥1, there exist constants ak,k0∈N, k0≤k≤2k0, such that
(169)τ∂τ2+∂τk0uτ=∑k=k02k0ak,k0τk-k0∂τkuτ
for all holomorphic functions u:Ω→C on an open set Ω⊂C.Lemma 25.
Leta,b,c≥0 be positive integers such that a≥b and a≥c. We put δ=a+b-c. Then, for all holomorphic functions u:Ω→C, the function ∂τ-a(τb∂τcu(τ)) can be written in the form
(170)∂τ-aτb∂τcuτ=∑(b′,c′)∈Oδαb′,c′τb′∂τc′uτ,
where Oδ is a finite subset of Z2 such that, for all (b′,c′)∈Oδ, b′-c′=δ, b′≥0, c′≤0, and αb′,c′∈Z.Using the latter Lemmas24 and 25 together with assumption (160), we can rewrite (167) in the form
(171)τr2+-i∂z+1r1∂xSV^τ,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵϵrk0-smmmmmmmi·∑(r′,p′)∈Os-k0αr′,p′τr′∂τ-p′∂zk1∂xk2V^τ,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵϵ-rl0+l1-1∂τ-l0V^τ,z,x,ϵ*l1,
where Os-k0 is a finite subset of N2 such that, for all r′,p′∈Os-k0, we have r′+p′=s-k0 and αr′,p′∈Z, for the given initial data
(172)∂xjV^τ,z,0,ϵ=∑β0,…,βl≥0Vβ0,…,βl,jτ,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!,mmmmmmmmmmmmmimimmii0≤j≤S-1.From assumption (160), we deduce that assumptions (119) and (122) of Proposition 20 are fulfilled for (171). Hence, from Proposition 20, we deduce the existence of a constant I>0 (independent of ϵ) such that if inequality (161) holds, then the formal series V^(τ,z,x,ϵ) solution of (171) and (172) defines a holomorphic function VUd(τ,z,x,ϵ) on the product Ud×Hρ1′×D(0,ρ1)×D(0,ϵ0)∖{0} which satisfies the next bound estimates: there exists a constant C18>0 such that
(173)VUdτ,z,x,ϵ≤C181+τ2ϵ2r-1exp⁡σϵrζbτ
for all (τ,z,x,ϵ)∈Ud×Hρ1′×D(0,ρ1)×D(0,ϵ0)∖{0}. Moreover, from the proof of Proposition 20 (especially formula (136)), we also get that each formal series V^(β0,…,βl,βl+1)(τ,ϵ) defines a holomorphic function VUd,(β0,…,βl,βl+1)(τ,ϵ) on (Ud∪D(0,ρ(β0,…,βl+1)))×D(0,ϵ0)∖{0} with the following estimates: there exist two constants C19>0, K19>0 such that
(174)VUd,β0,…,βl,βl+1τ,ϵ≤IC19exp⁡σϵrrbβ0,…,βl+1τl+2M0∑j=0lβj·K19βl+1β0!⋯βl+1!
for all τ∈D(0,ρ(β0,…,βl+1))∪Ud and ϵ∈D(0,ϵ0)∖{0}. From (174), we get that each formal series Y^(β0,…,βl,βl+1)(T,ϵ) is 1-summable with respect to the fact that T is the direction d and its 1-sum denoted by YUd,D(0,ϵ0)∖{0},(β0,…,βl,βl+1)(T,ϵ) can be written as Laplace transform of order 1 in the direction d of the function τ↦VUd,(β0,…,βl,βl+1)(τ,ϵ). Moreover, again by (174), we deduce that the series
(175)YUd,D0,ϵ0∖0T,z,x,ϵ=∑β0,…,βl,βl+1≥0YUd,D0,ϵ0∖0,β0,…,βl,βl+1T,ϵmmmmmmmm·exp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!
defines a bounded holomorphic function on Ud,θ,h′|ϵ|r×Hρ1′×D(0,ρ1) for some h′>0 and ρ1>0, for all ϵ∈D(0,ϵ0)∖{0}. Finally, from the algebraic properties of the κ-summation procedure (see [18] Section 6.3) since Y^(T,z,x,ϵ) formally solves (149), we deduce that YUd,D(0,ϵ0)∖{0}(T,z,x,ϵ) is an actual solution of the Cauchy problem (149), (150).


## 4. Formal Series Solutions and Gevrey Asymptotic Expansion in a Complex Parameter for the Main Cauchy Problem

### 4.1. Analytic Solutions in a Complex Parameter for the Main Cauchy Problem

We recall the definition of a good covering.Definition 26.
Letν≥2 be an integer. For all 0≤i≤ν-1, we consider an open sector Ei with vertex at 0 and with radius ϵ0. We assume that these sectors are three by three disjoint and that Ei+1∩Ei≠∅, for all 0≤i≤ν-1, where by convention we define Eν=E0. Moreover, we assume that ∪0≤i≤ν-1Ei=U∖{0}, where U is some neighborhood of 0 in C. Such a set of sectors {Ei}0≤i≤ν-1 is called a good covering in C*.Definition 27.
Let{Ei}0≤i≤ν-1 be a good covering in C*. Let r>0 be a positive real number and let r2≥1 be some integer. Let T be an open sector with vertex at 0 with radius rT>0. We consider the following family of open sectors:
(176)Udi,θ,ϵ0rrT=t∈C*:t<ϵ0rrT,di-argt<θ2,
where di∈R, 0≤i≤ν-1 and θ>π, which satisfy the following properties.(1)
For all0≤i≤ν-1, di≠π2k+1/r2 for all 0≤k≤r2-1.(2)
For all0≤i≤ν-1, all t∈T, and all ϵ∈Ei, we have that ϵrt∈Udi,θ,ϵ0rrT.
Under the above settings, we say that the family{{Udi,θ,ϵ0rrT}0≤i≤ν-1,T} is associated with the good covering {Ei}0≤i≤ν-1.LetS≥1 be an integer. Let S be a finite subset of N4, and let N be a finite subset of N2.As in the previous section, for all(s,k0,k1,k2)∈S and all integers β0,βl+1≥0, we denote by bs,k0,k1,k2,β0,βl+1(ϵ) some holomorphic function on D(0,ϵ0) which satisfies the next estimates: there exist constants ρ,ρ′>0, bs,k0,k1,k2>0 with
(177)sup⁡ϵ∈D(0,ϵ0)bs,k0,k1,k2,β0,βl+1ϵ≤bs,k0,k1,k2e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0. Likewise, for all (l0,l1)∈N and all integers β0,βl+1≥0, we denote by cl0,l1,β0,βl+1(ϵ) some holomorphic function on D(0,ϵ0) with the following estimates: there exists a constant cl0,l1>0 with
(178)sup⁡ϵ∈D(0,ϵ0)cl0,l1,β0,βl+1ϵ≤cl0,l1e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0.For all(s,k0,k1,k2)∈S and all (l0,l1)∈N, we consider the series
(179)bs,k0,k1,k2z,x,ϵ=∑β0,βl+1≥0bs,k0,k1,k2,β0,βl+1ϵeizβ0β0!xβl+1βl+1!,cl0,l1z,x,ϵ=∑β0,βl+1≥0cl0,l1,β0,βl+1ϵeizβ0β0!xβl+1βl+1!
which define bounded holomorphic functions on Hρ′×D(0,ρ)×D(0,ϵ0). Let {Ei}0≤i≤ν-1 be a good covering in C* and let r1,r2,r3≥1 be three integers. We put r=r3/r2.For all0≤i≤ν-1, we consider the following Cauchy problem:
(180)ϵr3t2∂t+tr2+-i∂z+1r1∂xSXit,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵts∂tk0∂zk1∂xk2Xit,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵtl0+l1-1Xit,z,x,ϵl1
for given initial data
(181)∂xjXit,z,0,ϵ=Ξi,jt,z,ϵ,0≤j≤S-1,
where the functions Ξi,j are constructed as follows. We consider a family of sectors {{Udi,θ,ϵ0rrT}0≤i≤ν-1,T} associated with the good covering {Ei}0≤i≤ν-1. For all 0≤i≤ν-1, let Udi be an unbounded open sector centered at 0, with bisecting direction di and with aperture ni>θ-π. We choose θ and ni in such a way that
(182)argτ≠π2k+1r2
for all τ∈Udi, all 0≤i≤ν-1, and all 0≤k≤r2-1. For all 0≤i≤ν-1 and all 0≤j≤S-1, we define
(183)Ξi,jt,z,ϵ=YUdi,D0,ϵ0∖0,jϵrt,z,ϵ,
where YUdi,D(0,ϵ0)∖{0},j(T,z,ϵ) is given by expression (158) and constructed as in the beginning of Section 3.2 with the help of a family of functions VUdi,(β0,…,βl,j)(τ,ϵ), for (β0,…,βl)∈Nl+1, 0≤j≤S-1, satisfying (152).We make the additional assumption that, for all(β0,…,βl)∈Nl+1, 0≤j≤S-1, there exists a holomorphic function τ↦V(β0,…,βl,j)(τ,ϵ) on D(0,ρ(β0,…,βl+1)) for all ϵ∈D(0,ϵ0)∖{0} such that
(184)VUdi,β0,…,βl,jτ,ϵ=Vβ0,…,βl,jτ,ϵ
for all 0≤i≤ν-1, all 0≤j≤S-1, all τ∈D(0,ρ(β0,…,βl+1)), and all ϵ∈D(0,ϵ0)∖{0}.Moreover, we assume that the seriesφdi,j(Z0,…,Zl,ϵ) defined in (154) belong to the Banach space G(Z¯00,…,Z¯l0,X¯0) for all ϵ∈D(0,ϵ0)∖{0}, where X¯0>0 and Z¯j0, 0≤j≤l, are chosen in such a way that Z¯j0>M0 for M0>0 that fulfills inequality (153) for some real number ρ1′>0.By construction,Ξi,j(t,z,ϵ) defines a holomorphic function on T×Hρ1′×Ei, for all 0≤i≤ν-1 and all 0≤j≤S-1, for well-chosen radius rT>0 and aperture θ.Proposition 28.
Let the initial data (181) be constructed as above. We make the following assumptions: for all (s,k0,k1,k2)∈S and all (l0,l1)∈N, we have that
(185)s≥2k0,S>k2,S>bs-k0+2+k2,S≥hr1+bs-k0+2+k1+k2,l1≥2.
Then, there exists a constant I>0 (independent of ϵ) such that if one assumes that
(186)φdi,jZ0,…,Zl,ϵZ¯00,…,Z¯l0,X¯0<I
for all 0≤i≤ν-1, for all ϵ∈D(0,ϵ0)∖{0}, the problem (180), (181) has a solution Xi(t,z,x,ϵ) which is holomorphic and bounded on (T∩D(0,h′))×Hρ1′×D(0,ρ1)×Ei, for some ρ1>0. Moreover, there exist constants K23,M23>0 and 0<h′′<h′ such that
(187)sup⁡t∈T∩D(0,h′′),z∈Hρ1′,x∈D(0,ρ1)Xi+1t,z,x,ϵ-Xit,z,x,ϵ≤K23exp⁡-M23ϵr3/hr1+r2
for all ϵ∈Ei+1∩Ei and all 0≤i≤ν-1 (where by convention Xν=X0), provided that ϵ0>0 is small enough.Proof. For all 0≤i≤ν-1, we consider the singular Cauchy problem (149) with initial data
(188)∂xjYUdi,D0,ϵ0∖0T,z,0,ϵ=YUdi,D0,ϵ0∖0,jT,z,ϵ,0≤j≤S-1.
Bearing in mind hypothesis (185) and assumption (186), we see that the assumptions of Proposition 23 are all fulfilled for the problem (149), (188), which, therefore, possesses a solution (T,z,x)↦YUdi,D(0,ϵ0)∖{0}(T,z,x,ϵ), holomorphic and bounded on Udi,θ,h′|ϵ|r×Hρ1′×D(0,ρ1) for some h′>0 and ρ1>0, for all ϵ∈D(0,ϵ0)∖{0}. Now, we put
(189)Xit,z,x,ϵ=YUdi,D0,ϵ0∖0ϵrt,z,x,ϵ
which defines a holomorphic and bounded function on (T∩D(0,h′))×Hρ1′×D(0,ρ1)×Ei, for all 0≤i≤ν-1, by construction of T and Ei in Definition 27. Since YUdi,D(0,ϵ0)∖{0}(T,z,x,ϵ) solves the problem (149), (188), one can check that Xi(t,z,x,ϵ) solves the problem (180), (181) on (T∩D(0,h′))×Hρ1′×D(0,ρ1)×Ei, for all 0≤i≤ν-1.In the next step of the proof, we show estimates (187). Let 0≤i≤ν-1. Using Proposition 23, we can write the function Xi(t,z,x,ϵ) as follows:
(190)Xit,z,x,ϵ=∑β0,…,βl,βl+1≥0Xi,β0,…,βl+1t,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!,
where
(191)Xi,β0,…,βl+1t,ϵ=1ϵrt∫LγiVUdi,β0,…,βl+1τ,ϵe-τ/ϵrtdτ
with integration path Lγi=R+e-1γi⊂Udi and such that τ↦VUdi,(β0,…,βl+1)(τ,ϵ) are holomorphic functions on D(0,ρ(β0,…,βl+1))∪Udi, for all ϵ∈D(0,ϵ0)∖{0}, and satisfy the estimates: there exist constants C20,K20>0, which satisfy ρ1K20<1, with
(192)VUdi,(β0,…,βl,βl+1)τ,ϵ≤C20exp⁡σϵrζbτl+2M0∑j=0lβjK20βl+1β0!⋯βl+1!
for all τ∈D(0,ρ(β0,…,βl,βl+1))∪Udi and ϵ∈D(0,ϵ0)∖{0}. Moreover, from assumption (184), we deduce with the help of recursion (107), which is satisfied for the coefficients of the formal solution V^(τ,z,x,ϵ) of the problem (171), (172), that, for all (β0,…,βl,βl+1)∈Nl+2, there exists a holomorphic function τ↦V(β0,…,βl,βl+1)(τ,ϵ) on D(0,ρ(β0,…,βl,βl+1)) for all ϵ∈D(0,ϵ0)∖{0} such that
(193)VUdi,β0,…,βl,βl+1τ,ϵ=Vβ0,…,βl,βl+1τ,ϵ
for all τ∈D(0,ρ(β0,…,βl,βl+1)), all ϵ∈D(0,ϵ0)∖{0}, and all 0≤i≤ν-1.We show the following.Lemma 29.
There exist constants0<h′′<h′, C21,K21,M21>0 (independent of ϵ), which satisfy ρ1K21<1, such that
(194)sup⁡t∈T∩D(0,h′′)Xi+1,β0,…,βl+1t,ϵ-Xi,β0,…,βl+1t,ϵ≤C21l+2M0∑j=0lβjK21βl+1β0!⋯βl+1!e-M21ρβ0,…,βl+1/ϵr
for all ϵ∈Ei+1∩Ei, all 0≤i≤ν-1, and all (β0,…,βl+1)∈Nl+2(where by convention Xν,(β0,…,βl+1)=X0,(β0,…,βl+1)).Proof.
From the fact that the functionτ↦V(β0,…,βl,βl+1)(τ,ϵ)e-τ/ϵrt is holomorphic on D(0,ρ(β0,…,βl,βl+1)), for all ϵ∈D(0,ϵ0), we deduce that its integral along the union of a segment starting from 0 to (ρ(β0,…,βl+1)/2)e-1γi+1, an arc of circle with radius ρ(β0,…,βl+1)/2 connecting (ρ(β0,…,βl+1)/2)e-1γi+1 and (ρ(β0,…,βl+1)/2)e-1γi and a segment starting from (ρ(β0,…,βl+1)/2)e-1γi to the origin, is vanishing. Therefore, using property (193), we can write, for all (β0,…,βl,βl+1)∈Nl+2,
(195)Xi+1,β0,…,βl+1t,ϵ-Xi,β0,…,βl+1t,ϵ=1ϵrt∫Lρβ0,…,βl+1/2,γi+1VUdi+1,β0,…,βl+1τ,ϵe-τ/ϵrtdτmmmiii-∫Lρβ0,…,βl+1/2,γiVUdi,β0,…,βl+1τ,ϵe-τ/ϵrtdτmmmiii+∫Cρβ0,…,βl+1/2,γi,γi+1Vβ0,…,βl+1τ,ϵe-τ/ϵrtdτ∫Lρβ0,…,βl+1/2,γi+1VUdi+1,β0,…,βl+1τ,ϵe-τ/ϵrtdτ,
where
(196)Lρβ0,…,βl+1/2,γi+1=ρβ0,…,βl+1/2,+∞e-1γi+1,Lρβ0,…,βl+1/2,γi=ρβ0,…,βl+1/2,+∞e-1γi
and C(ρ(β0,…,βl+1)/2,γi,γi+1) is an arc of circle centered at 0 with radius ρ(β0,…,βl+1)/2 connecting (ρ(β0,…,βl+1)/2)e-1γi and (ρ(β0,…,βl+1)/2)e-1γi+1 for a well-chosen orientation.
First, we give estimates for(197)I1=1ϵrt∫Lρβ0,…,βl+1/2,γi+1VUdi+1,β0,…,βl+1τ,ϵe-τ/ϵrtdτ.
By construction, the direction γi+1 (which depends on ϵrt) is chosen in such a way that cos⁡(γi+1-arg(ϵrt))≥δ1 for all ϵ∈Ei+1∩Ei and all t∈T∩D(0,h′), for some fixed δ1>0. From estimates (192), we deduce that
(198)I1≤1ϵrt∫ρβ0,…,βl+1/2+∞C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!mmmmm·eσζb/ϵrhe-h/ϵrtcos⁡γi+1-argϵrtdh≤1ϵrtC20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·∫ρβ0,…,βl+1/2+∞eσζb-δ1/th/ϵrdh=C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·1δ1-σζbte-δ1/t-σζbρβ0,…,βl+1/2ϵr≤C20δ2l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·e-δ2ρβ0,…,βl+1/2ϵrh′
for all t∈T∩D(0,h′) with |t|<(δ1-δ2)/σζ(b), for some 0<δ2<δ1 and for all ϵ∈Ei+1∩Ei.
Now, we provide estimates for(199)I2=1ϵrt∫Lρβ0,…,βl+1/2,γiVUdi,β0,…,βl+1τ,ϵe-τ/ϵrtdτ.
By construction, the direction γi (wich depends on ϵrt) is chosen in such a way that cos⁡(γi-arg(ϵrt))≥δ1 for all ϵ∈Ei+1∩Ei and all t∈T∩D(0,h′), for some fixed δ1>0. Again, from estimates (192), we deduce as above that
(200)I2≤C20δ2l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!e-δ2ρβ0,…,βl+1/2ϵrh′
for all t∈T∩D(0,h′) with t<(δ1-δ2)/σζ(b), for some 0<δ2<δ1 and for all ϵ∈Ei+1∩Ei.
Finally, we give upper bounds for(201)I3=∫Cρβ0,…,βl+1/2,γi,γi+1Vβ0,…,βl+1τ,ϵe-τ/ϵrtdτ.
Due to (192) and bearing in mind property (193), we deduce that
(202)I3≤1ϵrt∫γiγi+1C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!imm·eσζb/ϵrρβ0,…,βl+1/2imm·e-ρβ0,…,βl+1/2cos⁡θ-argϵrt/ϵrtimm·∫γiγi+1C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!ρβ0,…,βl+12dθ.
By construction, the circle C(ρ(β0,…,βl+1)/2,γi,γi+1) is chosen in order that cos⁡(θ-arg(ϵrt))≥δ1 for all θ∈[γi,γi+1] (if γi<γi+1), for all θ∈[γi+1,γi] (if γi+1<γi), and for every t∈T∩D(0,h′), ϵ∈Ei+1∩Ei. From (202), we get that
(203)I3≤γi+1-γiC20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·ρβ0,…,βl+121ϵrte-δ1/t-σζbρβ0,…,βl+1/2ϵr≤γi+1-γiC20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·ρβ0,…,βl+121ϵrte-δ2/4ρβ0,…,βl+1/ϵrt·e-δ2/4ρβ0,…,βl+1/ϵrh′
for all t∈T∩D(0,h′) with |t|<(δ1-δ2)/σζ(b), for some 0<δ2<δ1, and for all ϵ∈Ei+1∩Ei. Regarding inequality (70), we deduce from (203) that
(204)I3≤2e-1γi+1-γiδ2C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·e-δ2/4ρβ0,…,βl+1/ϵrh′
for all t∈T∩D(0,h′) with |t|<(δ1-δ2)/σζ(b), for some 0<δ2<δ1, and for all ϵ∈Ei+1∩Ei.
Finally, gathering decomposition (195) and estimates (198), (200), and (204), we get inequality (194).From Lemma29 and taking into account assumption (153), we can write
(205)sup⁡t∈T∩D(0,h′′),z∈Hρ1′,x∈D(0,ρ1)Xi+1t,z,x,ϵ-Xit,z,x,ϵ≤∑β0,…,βl,βl+1≥0C21e-M21ρβ0,…,βl+1/ϵrmmmmmmmm·l+2eρ1′max⁡j=0lξjM0∑j=0lβjρ1K21βl+1≤C21∑β0,…,βl,βl+1≥0e-M21ρβ0,…,βl+1/ϵrmmmmmmmmmmm·max⁡ρ1K21,12∑j=0l+1βj
for all ϵ∈Ei+1∩Ei and all 0≤i≤ν-1. Moreover, we recall that, for all integers l≥1, κ≥0,
(206)Cardβ0,…,βl+1∈Nl+2∑j=0l+1βj=κ=κ+l+1!l+1!κ!=Plκl+1!,
where Pl(κ)=(κ+l+1)(κ+l)⋯(κ+1) is a polynomial of degree l+1 in κ. From definition (89) and with the help of (206), we get two constants C22>0 and 0<K22<1 such that
(207)C21∑β0,…,βl,βl+1≥0e-M21ρβ0,…,βl+1/ϵrmax⁡ρ1K21,12∑j=0l+1βj≤C21∑κ≥0Plκl+1!exp⁡-M21Cξ1,…,ξlr1/r221+κhr1/r2ϵrmmmmmm·max⁡ρ1K21,12κ≤C22∑κ≥0exp⁡-M21Cξ1,…,ξlr1/r221+κhr1/r2ϵrK22κ
for all ϵ∈Ei+1∩Ei, all 0≤i≤ν-1. Now, we recall the following lemma from [4].Lemma 30.
Let0<a<1 and α>0. There exist K,M>0 and δ>0 such that
(208)∑κ≥0e-1/κ+1α1/ϵaκ≤Kexp⁡-Mϵ-1/α+1
for all ϵ∈(0,δ].From Lemma30 applied to inequality (207) and from (205), we deduce that estimates (187) hold, if ϵ0 is chosen small enough. Thus proof of Proposition 28 is complete.


### 4.2. Existence of Formal Power Series Solutions in the Complex Parameter for the Main Cauchy Problem

This subsection is devoted to explaining the main result of this work. Namely, we will establish the existence of a formal power series(209)X^t,z,x,ϵ=∑k≥0Hkt,z,xϵkk!∈OT∩D0,h′′×Hρ1′×D0,ρ1ϵ
which solves formally (180) and is constructed in such a way that the actual solutions Xi(t,z,x,ϵ) of the problem (180), (181) all have X^ as asymptotic expansion of Gevrey order hr1+r2/r3 on Ei (as formal series and functions with coefficients and values in the Banach space O((T∩D(0,h′′))×Hρ1′×D(0,ρ1)) equipped with the supremum norm) (see Definition 31 below), for all 0≤i≤ν-1.The proof makes use of a Banach valued version of cohomological criterion for Gevrey asymptotic expansion of sectorial holomorphic functions known in the literature as the Ramis-Sibuya theorem. For a reference, we refer to [18, Section 7.4 Proposition 18] and [30, Lemma XI-2-6].Definition 31.
Let(E,·E) be a complex Banach space over C. One considers a formal series G^(ϵ)=∑n≥0‍Gnϵn where the coefficients Gn belong to E and a holomorphic function G:E→E on an open bounded sector E with vertex at 0. Let s>0 be a positive real number. One says that G admits G^ as its asymptotic expansion of Gevrey order s on E if, for every proper and bounded subsector T of E, there exist two constants K,M>0 such that, for all N≥1, one has
(210)Gϵ-∑n=0N-1GnϵnE≤KMNN!sϵN
for all ϵ∈T.Theorem RS.Let(E,·E)be a complex Banach space overC. Let{Ei}0≤i≤ν-1be a good covering inC*. For every0≤i≤ν-1, letGibe a holomorphic function fromEiintoE, and let the cocycleΔi(ϵ):=Gi+1(ϵ)-Gi(ϵ)be a holomorphic function fromZi:=Ei∩Ei+1intoE(with the convention thatEν=E0andGν=G0). We assume that(1)
Gi(ϵ)is bounded asϵ∈Eitends to 0, for every0≤i≤ν-1;(2)
Δihas an exponential decreasing of orders>0onZi, for every0≤i≤ν-1, meaning that there existCi,Ai>0such that(211)ΔiϵE≤Cie-Ai/ϵ1/s,for everyϵ∈Ziand0≤i≤ν-1.Then, there exists a formal power series
G
^
(
ϵ
)
∈
E
[
[
ϵ
]
]
such that
G
i
(
ϵ
)
admits
G
^
(
ϵ
)
as its asymptotic expansion of Gevrey order
s
on
E
i
, for every
0
≤
i
≤
ν
-
1
.We now state the main result of our paper.Theorem 32.
Let one assume that conditions (185) hold. For all 0≤i≤ν-1, 0≤j≤S-1, one also assumes that for the functions Ξi,j(t,z,ϵ) constructed in (181) inequalities (186) are fulfilled for some constant I>0 given in Proposition 28. Let
(212)E=OT∩D0,h′′×Hρ1′×D0,ρ1
be the Banach space of holomorphic and bounded functions on (T∩D0,h′′)×Hρ1′×D(0,ρ1) equipped with the supremum norm, where h′′,ρ1′,ρ1 are the constants appearing in Proposition 28.
Then, there exists a formal series(213)X^t,z,x,ϵ=∑k≥0Hkt,z,xϵkk!∈Eϵ
which formally solves the equation
(214)ϵr3t2∂t+tr2+-i∂z+1r1∂xSX^t,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵts∂tk0∂zk1∂xk2X^t,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵtl0+l1-1X^t,z,x,ϵl1
and is the Gevrey asymptotic expansion of order hr1+r2/r3 of the E-valued function ϵ∈Ei↦Xi(t,z,x,ϵ), solution of the problem (180), (181) constructed in Proposition 28, for all 0≤i≤ν-1.Proof.
We consider the functionsXi(t,z,x,ϵ), 0≤i≤ν-1 constructed in Proposition 28. For all 0≤i≤ν-1, we define Gi(ϵ):=(t,z,x)↦Xi(t,z,x,ϵ), which is, by construction, a bounded holomorphic function from Ei into the Banach space E of holomorphic and bounded functions on (T∩D(0,h′′))×Hρ1′×D(0,ρ1) equipped with the supremum norm, where h′′,ρ1′,ρ1 are constants appearing in Proposition 28. Bearing in mind estimates (187), we deduce that the cocycle Δi(ϵ)=Gi+1(ϵ)-Gi(ϵ) fulfills estimates of form (211) on Zi=Ei+1∩Ei, where s=hr1+r2/r3, for all 0≤i≤ν-1. According to Theorem RS stated above, we deduce the existence of a formal series G^(ϵ)∈E[[ϵ]] which is the Gevrey asymptotic expansion of order hr1+r2/r3 of Gi(ϵ) on Ei, for all 0≤i≤ν-1. Let us define
(215)G^ϵ=X^t,z,x,ϵ=∑k≥0Hkt,z,xϵkk!.
It only remains to show that X^ is a formal solution of (214). From the fact that Gi(ϵ) admits G^(ϵ) as its asymptotic expansion at 0 on Ei, one gets
(216)lim⁡ϵ→0,ϵ∈Eisup⁡t∈T∩D(0,h′′),z∈Hρ1′,x∈D(0,ρ1)∂ϵlXit,z,x,ϵmmmmmmmmmmmmmmmm-∂ϵlHlt,z,x=0
for all l≥0 and all 0≤i≤ν-1. Now, we choose an integer i∈{0,…,ν-1}. By construction, the function Xi(t,z,x,ϵ) solves (180). We differentiate it l times with respect to ϵ. By means of Leibniz's rule, we get that ∂ϵlXi(t,z,x,ϵ) satisfies the identity
(217)∑h1+h2=ll!h1!h2!∂ϵh1ϵr3t2∂t+tr2∂xS∂ϵh2Xit,z,x,ϵhhhhhh+-i∂z+1r1∂xS∂ϵlXit,z,x,ϵ=∑(s,k0,k1,k2)∈Stsmmmmmm·∑h1+h2=ll!h1!h2!∂ϵh1bs,k0,k1,k2z,x,ϵmmmmmmmmmmm·∑h1+h2=ll!h1!h2!∂ϵh1bs,k0,k1,k2z,x,ϵ∂tk0∂zk1∂xk2∂ϵh2Xit,z,x,ϵ+∑(l0,l1)∈Ntl0+l1-1·∑h0+h1+⋯+hl1=ll!h0!⋯hl1!∂ϵh0cl0,l1z,x,ϵ·∏j=1l1∂ϵhjXit,z,x,ϵ
for all l≥1 and all (t,z,x,ϵ)∈(T∩D(0,h′′))×Hρ1′×D(0,ρ1)×Ei. We let ϵ tend to 0 in equality (217) and, with the help of (216), we get the following recursions:
(218)-i∂z+1r1∂xSHlt,z,xl!=∑(s,k0,k1,k2)∈Sts·∑h1+h2=l∂ϵh1bs,k0,k1,k2z,x,0h1!∂tk0∂zk1∂xk2Hh2t,z,xh2!+∑(l0,l1)∈Ntl0+l1-1·∑h0+h1+⋯+hl1=l∂ϵh0cl0,l1z,x,0h0!∏j=1l1Hhjt,z,xhj!
for all 0≤l<r3, all (t,z,x)∈(T∩D0,h′′)×Hρ1′×D(0,ρ1), and
(219)t2∂t+tr2∂xSHl-r3t,z,xl-r3!+-i∂z+1r1∂xSHlt,z,xl!=∑(s,k0,k1,k2)∈Sts·∑h1+h2=l∂ϵh1bs,k0,k1,k2z,x,0h1!∂tk0∂zk1∂xk2Hh2t,z,xh2!+∑(l0,l1)∈Ntl0+l1-1·∑h0+h1+⋯+hl1=l∂ϵh0cl0,l1z,x,0h0!∏j=1l1Hhjt,z,xhj!
for every l≥r3 and all (t,z,x)∈(T∩D(0,h′′))×Hρ1′×D(0,ρ1). Since the functions bs,k0,k1,k2(z,x,ϵ) and cl0,l1(z,x,ϵ) are analytic with respect to ϵ near the origin in C, we get that
(220)bs,k0,k1,k2z,x,ϵ=∑h≥0∂ϵhbs,k0,k1,k2z,x,0h!ϵhcl0,l1z,x,ϵ=∑h≥0∂ϵhcl0,l1z,x,0h!ϵh
for all (s,k0,k1,k2)∈S, (l0,l1)∈N and all (z,x,ϵ)∈Hρ1′×D(0,ρ1)×D(0,ϵ0). Finally, gathering recursions (218) and (219) and expansions (220), one can see that the formal series (215) solves (214).


## 4.1. Analytic Solutions in a Complex Parameter for the Main Cauchy Problem

We recall the definition of a good covering.Definition 26.
Letν≥2 be an integer. For all 0≤i≤ν-1, we consider an open sector Ei with vertex at 0 and with radius ϵ0. We assume that these sectors are three by three disjoint and that Ei+1∩Ei≠∅, for all 0≤i≤ν-1, where by convention we define Eν=E0. Moreover, we assume that ∪0≤i≤ν-1Ei=U∖{0}, where U is some neighborhood of 0 in C. Such a set of sectors {Ei}0≤i≤ν-1 is called a good covering in C*.Definition 27.
Let{Ei}0≤i≤ν-1 be a good covering in C*. Let r>0 be a positive real number and let r2≥1 be some integer. Let T be an open sector with vertex at 0 with radius rT>0. We consider the following family of open sectors:
(176)Udi,θ,ϵ0rrT=t∈C*:t<ϵ0rrT,di-argt<θ2,
where di∈R, 0≤i≤ν-1 and θ>π, which satisfy the following properties.(1)
For all0≤i≤ν-1, di≠π2k+1/r2 for all 0≤k≤r2-1.(2)
For all0≤i≤ν-1, all t∈T, and all ϵ∈Ei, we have that ϵrt∈Udi,θ,ϵ0rrT.
Under the above settings, we say that the family{{Udi,θ,ϵ0rrT}0≤i≤ν-1,T} is associated with the good covering {Ei}0≤i≤ν-1.LetS≥1 be an integer. Let S be a finite subset of N4, and let N be a finite subset of N2.As in the previous section, for all(s,k0,k1,k2)∈S and all integers β0,βl+1≥0, we denote by bs,k0,k1,k2,β0,βl+1(ϵ) some holomorphic function on D(0,ϵ0) which satisfies the next estimates: there exist constants ρ,ρ′>0, bs,k0,k1,k2>0 with
(177)sup⁡ϵ∈D(0,ϵ0)bs,k0,k1,k2,β0,βl+1ϵ≤bs,k0,k1,k2e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0. Likewise, for all (l0,l1)∈N and all integers β0,βl+1≥0, we denote by cl0,l1,β0,βl+1(ϵ) some holomorphic function on D(0,ϵ0) with the following estimates: there exists a constant cl0,l1>0 with
(178)sup⁡ϵ∈D(0,ϵ0)cl0,l1,β0,βl+1ϵ≤cl0,l1e-ρ′2β012ρβl+1β0!βl+1!
for all β0,βl+1≥0.For all(s,k0,k1,k2)∈S and all (l0,l1)∈N, we consider the series
(179)bs,k0,k1,k2z,x,ϵ=∑β0,βl+1≥0bs,k0,k1,k2,β0,βl+1ϵeizβ0β0!xβl+1βl+1!,cl0,l1z,x,ϵ=∑β0,βl+1≥0cl0,l1,β0,βl+1ϵeizβ0β0!xβl+1βl+1!
which define bounded holomorphic functions on Hρ′×D(0,ρ)×D(0,ϵ0). Let {Ei}0≤i≤ν-1 be a good covering in C* and let r1,r2,r3≥1 be three integers. We put r=r3/r2.For all0≤i≤ν-1, we consider the following Cauchy problem:
(180)ϵr3t2∂t+tr2+-i∂z+1r1∂xSXit,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵts∂tk0∂zk1∂xk2Xit,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵtl0+l1-1Xit,z,x,ϵl1
for given initial data
(181)∂xjXit,z,0,ϵ=Ξi,jt,z,ϵ,0≤j≤S-1,
where the functions Ξi,j are constructed as follows. We consider a family of sectors {{Udi,θ,ϵ0rrT}0≤i≤ν-1,T} associated with the good covering {Ei}0≤i≤ν-1. For all 0≤i≤ν-1, let Udi be an unbounded open sector centered at 0, with bisecting direction di and with aperture ni>θ-π. We choose θ and ni in such a way that
(182)argτ≠π2k+1r2
for all τ∈Udi, all 0≤i≤ν-1, and all 0≤k≤r2-1. For all 0≤i≤ν-1 and all 0≤j≤S-1, we define
(183)Ξi,jt,z,ϵ=YUdi,D0,ϵ0∖0,jϵrt,z,ϵ,
where YUdi,D(0,ϵ0)∖{0},j(T,z,ϵ) is given by expression (158) and constructed as in the beginning of Section 3.2 with the help of a family of functions VUdi,(β0,…,βl,j)(τ,ϵ), for (β0,…,βl)∈Nl+1, 0≤j≤S-1, satisfying (152).We make the additional assumption that, for all(β0,…,βl)∈Nl+1, 0≤j≤S-1, there exists a holomorphic function τ↦V(β0,…,βl,j)(τ,ϵ) on D(0,ρ(β0,…,βl+1)) for all ϵ∈D(0,ϵ0)∖{0} such that
(184)VUdi,β0,…,βl,jτ,ϵ=Vβ0,…,βl,jτ,ϵ
for all 0≤i≤ν-1, all 0≤j≤S-1, all τ∈D(0,ρ(β0,…,βl+1)), and all ϵ∈D(0,ϵ0)∖{0}.Moreover, we assume that the seriesφdi,j(Z0,…,Zl,ϵ) defined in (154) belong to the Banach space G(Z¯00,…,Z¯l0,X¯0) for all ϵ∈D(0,ϵ0)∖{0}, where X¯0>0 and Z¯j0, 0≤j≤l, are chosen in such a way that Z¯j0>M0 for M0>0 that fulfills inequality (153) for some real number ρ1′>0.By construction,Ξi,j(t,z,ϵ) defines a holomorphic function on T×Hρ1′×Ei, for all 0≤i≤ν-1 and all 0≤j≤S-1, for well-chosen radius rT>0 and aperture θ.Proposition 28.
Let the initial data (181) be constructed as above. We make the following assumptions: for all (s,k0,k1,k2)∈S and all (l0,l1)∈N, we have that
(185)s≥2k0,S>k2,S>bs-k0+2+k2,S≥hr1+bs-k0+2+k1+k2,l1≥2.
Then, there exists a constant I>0 (independent of ϵ) such that if one assumes that
(186)φdi,jZ0,…,Zl,ϵZ¯00,…,Z¯l0,X¯0<I
for all 0≤i≤ν-1, for all ϵ∈D(0,ϵ0)∖{0}, the problem (180), (181) has a solution Xi(t,z,x,ϵ) which is holomorphic and bounded on (T∩D(0,h′))×Hρ1′×D(0,ρ1)×Ei, for some ρ1>0. Moreover, there exist constants K23,M23>0 and 0<h′′<h′ such that
(187)sup⁡t∈T∩D(0,h′′),z∈Hρ1′,x∈D(0,ρ1)Xi+1t,z,x,ϵ-Xit,z,x,ϵ≤K23exp⁡-M23ϵr3/hr1+r2
for all ϵ∈Ei+1∩Ei and all 0≤i≤ν-1 (where by convention Xν=X0), provided that ϵ0>0 is small enough.Proof. For all 0≤i≤ν-1, we consider the singular Cauchy problem (149) with initial data
(188)∂xjYUdi,D0,ϵ0∖0T,z,0,ϵ=YUdi,D0,ϵ0∖0,jT,z,ϵ,0≤j≤S-1.
Bearing in mind hypothesis (185) and assumption (186), we see that the assumptions of Proposition 23 are all fulfilled for the problem (149), (188), which, therefore, possesses a solution (T,z,x)↦YUdi,D(0,ϵ0)∖{0}(T,z,x,ϵ), holomorphic and bounded on Udi,θ,h′|ϵ|r×Hρ1′×D(0,ρ1) for some h′>0 and ρ1>0, for all ϵ∈D(0,ϵ0)∖{0}. Now, we put
(189)Xit,z,x,ϵ=YUdi,D0,ϵ0∖0ϵrt,z,x,ϵ
which defines a holomorphic and bounded function on (T∩D(0,h′))×Hρ1′×D(0,ρ1)×Ei, for all 0≤i≤ν-1, by construction of T and Ei in Definition 27. Since YUdi,D(0,ϵ0)∖{0}(T,z,x,ϵ) solves the problem (149), (188), one can check that Xi(t,z,x,ϵ) solves the problem (180), (181) on (T∩D(0,h′))×Hρ1′×D(0,ρ1)×Ei, for all 0≤i≤ν-1.In the next step of the proof, we show estimates (187). Let 0≤i≤ν-1. Using Proposition 23, we can write the function Xi(t,z,x,ϵ) as follows:
(190)Xit,z,x,ϵ=∑β0,…,βl,βl+1≥0Xi,β0,…,βl+1t,ϵexp⁡iz∑j=0lβjξjβ0!⋯βl!xβl+1βl+1!,
where
(191)Xi,β0,…,βl+1t,ϵ=1ϵrt∫LγiVUdi,β0,…,βl+1τ,ϵe-τ/ϵrtdτ
with integration path Lγi=R+e-1γi⊂Udi and such that τ↦VUdi,(β0,…,βl+1)(τ,ϵ) are holomorphic functions on D(0,ρ(β0,…,βl+1))∪Udi, for all ϵ∈D(0,ϵ0)∖{0}, and satisfy the estimates: there exist constants C20,K20>0, which satisfy ρ1K20<1, with
(192)VUdi,(β0,…,βl,βl+1)τ,ϵ≤C20exp⁡σϵrζbτl+2M0∑j=0lβjK20βl+1β0!⋯βl+1!
for all τ∈D(0,ρ(β0,…,βl,βl+1))∪Udi and ϵ∈D(0,ϵ0)∖{0}. Moreover, from assumption (184), we deduce with the help of recursion (107), which is satisfied for the coefficients of the formal solution V^(τ,z,x,ϵ) of the problem (171), (172), that, for all (β0,…,βl,βl+1)∈Nl+2, there exists a holomorphic function τ↦V(β0,…,βl,βl+1)(τ,ϵ) on D(0,ρ(β0,…,βl,βl+1)) for all ϵ∈D(0,ϵ0)∖{0} such that
(193)VUdi,β0,…,βl,βl+1τ,ϵ=Vβ0,…,βl,βl+1τ,ϵ
for all τ∈D(0,ρ(β0,…,βl,βl+1)), all ϵ∈D(0,ϵ0)∖{0}, and all 0≤i≤ν-1.We show the following.Lemma 29.
There exist constants0<h′′<h′, C21,K21,M21>0 (independent of ϵ), which satisfy ρ1K21<1, such that
(194)sup⁡t∈T∩D(0,h′′)Xi+1,β0,…,βl+1t,ϵ-Xi,β0,…,βl+1t,ϵ≤C21l+2M0∑j=0lβjK21βl+1β0!⋯βl+1!e-M21ρβ0,…,βl+1/ϵr
for all ϵ∈Ei+1∩Ei, all 0≤i≤ν-1, and all (β0,…,βl+1)∈Nl+2(where by convention Xν,(β0,…,βl+1)=X0,(β0,…,βl+1)).Proof.
From the fact that the functionτ↦V(β0,…,βl,βl+1)(τ,ϵ)e-τ/ϵrt is holomorphic on D(0,ρ(β0,…,βl,βl+1)), for all ϵ∈D(0,ϵ0), we deduce that its integral along the union of a segment starting from 0 to (ρ(β0,…,βl+1)/2)e-1γi+1, an arc of circle with radius ρ(β0,…,βl+1)/2 connecting (ρ(β0,…,βl+1)/2)e-1γi+1 and (ρ(β0,…,βl+1)/2)e-1γi and a segment starting from (ρ(β0,…,βl+1)/2)e-1γi to the origin, is vanishing. Therefore, using property (193), we can write, for all (β0,…,βl,βl+1)∈Nl+2,
(195)Xi+1,β0,…,βl+1t,ϵ-Xi,β0,…,βl+1t,ϵ=1ϵrt∫Lρβ0,…,βl+1/2,γi+1VUdi+1,β0,…,βl+1τ,ϵe-τ/ϵrtdτmmmiii-∫Lρβ0,…,βl+1/2,γiVUdi,β0,…,βl+1τ,ϵe-τ/ϵrtdτmmmiii+∫Cρβ0,…,βl+1/2,γi,γi+1Vβ0,…,βl+1τ,ϵe-τ/ϵrtdτ∫Lρβ0,…,βl+1/2,γi+1VUdi+1,β0,…,βl+1τ,ϵe-τ/ϵrtdτ,
where
(196)Lρβ0,…,βl+1/2,γi+1=ρβ0,…,βl+1/2,+∞e-1γi+1,Lρβ0,…,βl+1/2,γi=ρβ0,…,βl+1/2,+∞e-1γi
and C(ρ(β0,…,βl+1)/2,γi,γi+1) is an arc of circle centered at 0 with radius ρ(β0,…,βl+1)/2 connecting (ρ(β0,…,βl+1)/2)e-1γi and (ρ(β0,…,βl+1)/2)e-1γi+1 for a well-chosen orientation.
First, we give estimates for(197)I1=1ϵrt∫Lρβ0,…,βl+1/2,γi+1VUdi+1,β0,…,βl+1τ,ϵe-τ/ϵrtdτ.
By construction, the direction γi+1 (which depends on ϵrt) is chosen in such a way that cos⁡(γi+1-arg(ϵrt))≥δ1 for all ϵ∈Ei+1∩Ei and all t∈T∩D(0,h′), for some fixed δ1>0. From estimates (192), we deduce that
(198)I1≤1ϵrt∫ρβ0,…,βl+1/2+∞C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!mmmmm·eσζb/ϵrhe-h/ϵrtcos⁡γi+1-argϵrtdh≤1ϵrtC20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·∫ρβ0,…,βl+1/2+∞eσζb-δ1/th/ϵrdh=C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·1δ1-σζbte-δ1/t-σζbρβ0,…,βl+1/2ϵr≤C20δ2l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·e-δ2ρβ0,…,βl+1/2ϵrh′
for all t∈T∩D(0,h′) with |t|<(δ1-δ2)/σζ(b), for some 0<δ2<δ1 and for all ϵ∈Ei+1∩Ei.
Now, we provide estimates for(199)I2=1ϵrt∫Lρβ0,…,βl+1/2,γiVUdi,β0,…,βl+1τ,ϵe-τ/ϵrtdτ.
By construction, the direction γi (wich depends on ϵrt) is chosen in such a way that cos⁡(γi-arg(ϵrt))≥δ1 for all ϵ∈Ei+1∩Ei and all t∈T∩D(0,h′), for some fixed δ1>0. Again, from estimates (192), we deduce as above that
(200)I2≤C20δ2l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!e-δ2ρβ0,…,βl+1/2ϵrh′
for all t∈T∩D(0,h′) with t<(δ1-δ2)/σζ(b), for some 0<δ2<δ1 and for all ϵ∈Ei+1∩Ei.
Finally, we give upper bounds for(201)I3=∫Cρβ0,…,βl+1/2,γi,γi+1Vβ0,…,βl+1τ,ϵe-τ/ϵrtdτ.
Due to (192) and bearing in mind property (193), we deduce that
(202)I3≤1ϵrt∫γiγi+1C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!imm·eσζb/ϵrρβ0,…,βl+1/2imm·e-ρβ0,…,βl+1/2cos⁡θ-argϵrt/ϵrtimm·∫γiγi+1C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!ρβ0,…,βl+12dθ.
By construction, the circle C(ρ(β0,…,βl+1)/2,γi,γi+1) is chosen in order that cos⁡(θ-arg(ϵrt))≥δ1 for all θ∈[γi,γi+1] (if γi<γi+1), for all θ∈[γi+1,γi] (if γi+1<γi), and for every t∈T∩D(0,h′), ϵ∈Ei+1∩Ei. From (202), we get that
(203)I3≤γi+1-γiC20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·ρβ0,…,βl+121ϵrte-δ1/t-σζbρβ0,…,βl+1/2ϵr≤γi+1-γiC20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·ρβ0,…,βl+121ϵrte-δ2/4ρβ0,…,βl+1/ϵrt·e-δ2/4ρβ0,…,βl+1/ϵrh′
for all t∈T∩D(0,h′) with |t|<(δ1-δ2)/σζ(b), for some 0<δ2<δ1, and for all ϵ∈Ei+1∩Ei. Regarding inequality (70), we deduce from (203) that
(204)I3≤2e-1γi+1-γiδ2C20l+2M0∑j=0lβjK20βl+1β0!⋯βl+1!·e-δ2/4ρβ0,…,βl+1/ϵrh′
for all t∈T∩D(0,h′) with |t|<(δ1-δ2)/σζ(b), for some 0<δ2<δ1, and for all ϵ∈Ei+1∩Ei.
Finally, gathering decomposition (195) and estimates (198), (200), and (204), we get inequality (194).From Lemma29 and taking into account assumption (153), we can write
(205)sup⁡t∈T∩D(0,h′′),z∈Hρ1′,x∈D(0,ρ1)Xi+1t,z,x,ϵ-Xit,z,x,ϵ≤∑β0,…,βl,βl+1≥0C21e-M21ρβ0,…,βl+1/ϵrmmmmmmmm·l+2eρ1′max⁡j=0lξjM0∑j=0lβjρ1K21βl+1≤C21∑β0,…,βl,βl+1≥0e-M21ρβ0,…,βl+1/ϵrmmmmmmmmmmm·max⁡ρ1K21,12∑j=0l+1βj
for all ϵ∈Ei+1∩Ei and all 0≤i≤ν-1. Moreover, we recall that, for all integers l≥1, κ≥0,
(206)Cardβ0,…,βl+1∈Nl+2∑j=0l+1βj=κ=κ+l+1!l+1!κ!=Plκl+1!,
where Pl(κ)=(κ+l+1)(κ+l)⋯(κ+1) is a polynomial of degree l+1 in κ. From definition (89) and with the help of (206), we get two constants C22>0 and 0<K22<1 such that
(207)C21∑β0,…,βl,βl+1≥0e-M21ρβ0,…,βl+1/ϵrmax⁡ρ1K21,12∑j=0l+1βj≤C21∑κ≥0Plκl+1!exp⁡-M21Cξ1,…,ξlr1/r221+κhr1/r2ϵrmmmmmm·max⁡ρ1K21,12κ≤C22∑κ≥0exp⁡-M21Cξ1,…,ξlr1/r221+κhr1/r2ϵrK22κ
for all ϵ∈Ei+1∩Ei, all 0≤i≤ν-1. Now, we recall the following lemma from [4].Lemma 30.
Let0<a<1 and α>0. There exist K,M>0 and δ>0 such that
(208)∑κ≥0e-1/κ+1α1/ϵaκ≤Kexp⁡-Mϵ-1/α+1
for all ϵ∈(0,δ].From Lemma30 applied to inequality (207) and from (205), we deduce that estimates (187) hold, if ϵ0 is chosen small enough. Thus proof of Proposition 28 is complete.


## 4.2. Existence of Formal Power Series Solutions in the Complex Parameter for the Main Cauchy Problem

This subsection is devoted to explaining the main result of this work. Namely, we will establish the existence of a formal power series(209)X^t,z,x,ϵ=∑k≥0Hkt,z,xϵkk!∈OT∩D0,h′′×Hρ1′×D0,ρ1ϵ
which solves formally (180) and is constructed in such a way that the actual solutions Xi(t,z,x,ϵ) of the problem (180), (181) all have X^ as asymptotic expansion of Gevrey order hr1+r2/r3 on Ei (as formal series and functions with coefficients and values in the Banach space O((T∩D(0,h′′))×Hρ1′×D(0,ρ1)) equipped with the supremum norm) (see Definition 31 below), for all 0≤i≤ν-1.The proof makes use of a Banach valued version of cohomological criterion for Gevrey asymptotic expansion of sectorial holomorphic functions known in the literature as the Ramis-Sibuya theorem. For a reference, we refer to [18, Section 7.4 Proposition 18] and [30, Lemma XI-2-6].Definition 31.
Let(E,·E) be a complex Banach space over C. One considers a formal series G^(ϵ)=∑n≥0‍Gnϵn where the coefficients Gn belong to E and a holomorphic function G:E→E on an open bounded sector E with vertex at 0. Let s>0 be a positive real number. One says that G admits G^ as its asymptotic expansion of Gevrey order s on E if, for every proper and bounded subsector T of E, there exist two constants K,M>0 such that, for all N≥1, one has
(210)Gϵ-∑n=0N-1GnϵnE≤KMNN!sϵN
for all ϵ∈T.Theorem RS.Let(E,·E)be a complex Banach space overC. Let{Ei}0≤i≤ν-1be a good covering inC*. For every0≤i≤ν-1, letGibe a holomorphic function fromEiintoE, and let the cocycleΔi(ϵ):=Gi+1(ϵ)-Gi(ϵ)be a holomorphic function fromZi:=Ei∩Ei+1intoE(with the convention thatEν=E0andGν=G0). We assume that(1)
Gi(ϵ)is bounded asϵ∈Eitends to 0, for every0≤i≤ν-1;(2)
Δihas an exponential decreasing of orders>0onZi, for every0≤i≤ν-1, meaning that there existCi,Ai>0such that(211)ΔiϵE≤Cie-Ai/ϵ1/s,for everyϵ∈Ziand0≤i≤ν-1.Then, there exists a formal power series
G
^
(
ϵ
)
∈
E
[
[
ϵ
]
]
such that
G
i
(
ϵ
)
admits
G
^
(
ϵ
)
as its asymptotic expansion of Gevrey order
s
on
E
i
, for every
0
≤
i
≤
ν
-
1
.We now state the main result of our paper.Theorem 32.
Let one assume that conditions (185) hold. For all 0≤i≤ν-1, 0≤j≤S-1, one also assumes that for the functions Ξi,j(t,z,ϵ) constructed in (181) inequalities (186) are fulfilled for some constant I>0 given in Proposition 28. Let
(212)E=OT∩D0,h′′×Hρ1′×D0,ρ1
be the Banach space of holomorphic and bounded functions on (T∩D0,h′′)×Hρ1′×D(0,ρ1) equipped with the supremum norm, where h′′,ρ1′,ρ1 are the constants appearing in Proposition 28.
Then, there exists a formal series(213)X^t,z,x,ϵ=∑k≥0Hkt,z,xϵkk!∈Eϵ
which formally solves the equation
(214)ϵr3t2∂t+tr2+-i∂z+1r1∂xSX^t,z,x,ϵ=∑(s,k0,k1,k2)∈Sbs,k0,k1,k2z,x,ϵts∂tk0∂zk1∂xk2X^t,z,x,ϵ+∑(l0,l1)∈Ncl0,l1z,x,ϵtl0+l1-1X^t,z,x,ϵl1
and is the Gevrey asymptotic expansion of order hr1+r2/r3 of the E-valued function ϵ∈Ei↦Xi(t,z,x,ϵ), solution of the problem (180), (181) constructed in Proposition 28, for all 0≤i≤ν-1.Proof.
We consider the functionsXi(t,z,x,ϵ), 0≤i≤ν-1 constructed in Proposition 28. For all 0≤i≤ν-1, we define Gi(ϵ):=(t,z,x)↦Xi(t,z,x,ϵ), which is, by construction, a bounded holomorphic function from Ei into the Banach space E of holomorphic and bounded functions on (T∩D(0,h′′))×Hρ1′×D(0,ρ1) equipped with the supremum norm, where h′′,ρ1′,ρ1 are constants appearing in Proposition 28. Bearing in mind estimates (187), we deduce that the cocycle Δi(ϵ)=Gi+1(ϵ)-Gi(ϵ) fulfills estimates of form (211) on Zi=Ei+1∩Ei, where s=hr1+r2/r3, for all 0≤i≤ν-1. According to Theorem RS stated above, we deduce the existence of a formal series G^(ϵ)∈E[[ϵ]] which is the Gevrey asymptotic expansion of order hr1+r2/r3 of Gi(ϵ) on Ei, for all 0≤i≤ν-1. Let us define
(215)G^ϵ=X^t,z,x,ϵ=∑k≥0Hkt,z,xϵkk!.
It only remains to show that X^ is a formal solution of (214). From the fact that Gi(ϵ) admits G^(ϵ) as its asymptotic expansion at 0 on Ei, one gets
(216)lim⁡ϵ→0,ϵ∈Eisup⁡t∈T∩D(0,h′′),z∈Hρ1′,x∈D(0,ρ1)∂ϵlXit,z,x,ϵmmmmmmmmmmmmmmmm-∂ϵlHlt,z,x=0
for all l≥0 and all 0≤i≤ν-1. Now, we choose an integer i∈{0,…,ν-1}. By construction, the function Xi(t,z,x,ϵ) solves (180). We differentiate it l times with respect to ϵ. By means of Leibniz's rule, we get that ∂ϵlXi(t,z,x,ϵ) satisfies the identity
(217)∑h1+h2=ll!h1!h2!∂ϵh1ϵr3t2∂t+tr2∂xS∂ϵh2Xit,z,x,ϵhhhhhh+-i∂z+1r1∂xS∂ϵlXit,z,x,ϵ=∑(s,k0,k1,k2)∈Stsmmmmmm·∑h1+h2=ll!h1!h2!∂ϵh1bs,k0,k1,k2z,x,ϵmmmmmmmmmmm·∑h1+h2=ll!h1!h2!∂ϵh1bs,k0,k1,k2z,x,ϵ∂tk0∂zk1∂xk2∂ϵh2Xit,z,x,ϵ+∑(l0,l1)∈Ntl0+l1-1·∑h0+h1+⋯+hl1=ll!h0!⋯hl1!∂ϵh0cl0,l1z,x,ϵ·∏j=1l1∂ϵhjXit,z,x,ϵ
for all l≥1 and all (t,z,x,ϵ)∈(T∩D(0,h′′))×Hρ1′×D(0,ρ1)×Ei. We let ϵ tend to 0 in equality (217) and, with the help of (216), we get the following recursions:
(218)-i∂z+1r1∂xSHlt,z,xl!=∑(s,k0,k1,k2)∈Sts·∑h1+h2=l∂ϵh1bs,k0,k1,k2z,x,0h1!∂tk0∂zk1∂xk2Hh2t,z,xh2!+∑(l0,l1)∈Ntl0+l1-1·∑h0+h1+⋯+hl1=l∂ϵh0cl0,l1z,x,0h0!∏j=1l1Hhjt,z,xhj!
for all 0≤l<r3, all (t,z,x)∈(T∩D0,h′′)×Hρ1′×D(0,ρ1), and
(219)t2∂t+tr2∂xSHl-r3t,z,xl-r3!+-i∂z+1r1∂xSHlt,z,xl!=∑(s,k0,k1,k2)∈Sts·∑h1+h2=l∂ϵh1bs,k0,k1,k2z,x,0h1!∂tk0∂zk1∂xk2Hh2t,z,xh2!+∑(l0,l1)∈Ntl0+l1-1·∑h0+h1+⋯+hl1=l∂ϵh0cl0,l1z,x,0h0!∏j=1l1Hhjt,z,xhj!
for every l≥r3 and all (t,z,x)∈(T∩D(0,h′′))×Hρ1′×D(0,ρ1). Since the functions bs,k0,k1,k2(z,x,ϵ) and cl0,l1(z,x,ϵ) are analytic with respect to ϵ near the origin in C, we get that
(220)bs,k0,k1,k2z,x,ϵ=∑h≥0∂ϵhbs,k0,k1,k2z,x,0h!ϵhcl0,l1z,x,ϵ=∑h≥0∂ϵhcl0,l1z,x,0h!ϵh
for all (s,k0,k1,k2)∈S, (l0,l1)∈N and all (z,x,ϵ)∈Hρ1′×D(0,ρ1)×D(0,ϵ0). Finally, gathering recursions (218) and (219) and expansions (220), one can see that the formal series (215) solves (214).



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*Source: 153169-2014-12-22.xml*