# On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

Authors: Stéphane Malek
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameterϵ. The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in ϵ of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in ϵ as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.


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## Body

## 1. Introduction

In this work, we consider a family of nonlinear singularly perturbed equations of the form(1)Q∂zP1t,ϵut,z,ϵ+P2t,ϵu2t,z,ϵ=ft,z,ϵ+P3t,ϵ,∂t,∂zut,z,ϵ,where Q, P1, P2, and P3 are polynomials with complex coefficients and f is an analytic function in the vicinity of the origin with respect to t and ϵ in C and holomorphic with respect to z on a horizontal strip in C of the form Hβ={z∈C∣Imz<β} for some β>0.Here we consider the case whenP1(0,ϵ) vanishes identically near 0. The point t=0 is known to be called aturning point in that situation; see [1, 2] for a more detailed description of this terminology in the linear and nonlinear settings. Let us recall the definition of the valuation valt(f) of an analytic function near t=0 as the smallest integer k≥0 with the factorization f(t)=tkf~(t) for an analytic function f~ near t=0 with f~(0)≠0. The most interesting case examined in this work is when the valuation valt(P1) of P1(t,ϵ) with respect to t is larger than the valuation valt(P2) or valt(f(t,z,ϵ)) since the problem cannot be reduced to the case P1(0,0)≠0 by dividing (1) by a suitable power of t and ϵ; see Remark 13.In our previous study [3], we already have considered a similar problem which corresponds to the situation when P1(0,0)≠0 for our equation (1). Namely, we focused on the following problem:(2)Q∂z∂tyt,z,ϵ=Q1∂zyt,z,ϵQ2∂zyt,z,ϵ+Ht,ϵ,∂t,∂zyt,z,ϵ+ft,z,ϵfor given vanishing initial data y(0,z,ϵ)≡0, where Q, Q1, Q2, and H are polynomials with complex coefficients and f(t,z,ϵ) is a forcing term constructed as above. Under appropriate assumptions on the shape of (2), we established the existence of a family of actual bounded holomorphic solutions yp(t,z,ϵ), 0≤p≤ς-1, for some integer ς≥2, defined on domains T×Hβ×Ep, for some fixed bounded sector T with vertex at 0 and E={Ep}0≤p≤ς-1, a set of bounded sectors whose union covers a full neighborhood of 0 in C∗. These solutions are obtained by means of Laplace and inverse Fourier transforms. On each sector Ep, they share with respect to ϵ a common asymptotic expansion y^(t,z,ϵ)=∑n≥0yn(t,z)ϵn which defines a formal series with bounded holomorphic coefficients on T×Hβ. Moreover, this asymptotic expansion is shown to be of Gevrey order (at most) 1/k that appears in the highest order term of the operator H which is of irregular type in the sense of [4] outlined as ϵ(δD-1)kt(δD-1)(k+1)∂tδDRD(∂z), for some integer δD≥2 and a polynomial RD with complex coefficients. Conjointly, since the aperture of the sectors Ep can be chosen slightly larger than π/k, the functions ϵ↦yp(t,z,ϵ) can be viewed as k-sums of the formal series y^ as defined in [5].In this work, our goal is to achieve a similar statement, namely, the existence of sectorial holomorphic solutions and asymptotic expansions asϵ tends to 0. However, the main contrast with problem (2) is that, due to the presence of the turning point, our solutions are no longer bounded in the vicinity of the origin, being meromorphic in both time variable t and parameter ϵ. Namely, we build a set of actual meromorphic solutions udp(t,z,ϵ) to problem (1) of the form (3)udpt,z,ϵ=ϵβU0ϵαt+ϵαtγvdpt,z,ϵ,where α>1, β are some rational numbers, γ is an integer, and U0(T) is a nonidentically vanishing root of a second-order algebraic equation with polynomial coefficients related to the polynomials P1, P2, see (70), and where vdp(t,z,ϵ) is a bounded holomorphic function on products T×Hβ×Ep similar to the ones mentioned above, which can be expressed as a Laplace transform of some order κ≥1 and Fourier inverse transform (4)vdpt,z,ϵ=κ2π1/2∫-∞+∞∫Ldpωκdpu,m,ϵexp⁡-uϵχ+αtκeizmduudmalong some half line Ldp=R+eidp, for some positive rational number χ>0, where ωκdp(u,m,ϵ) represents a function with at most exponential growth of order κ on a sector containing Ldp with respect to u, with exponential decay with respect to m on R and with analytic dependence on ϵ near 0 (see Theorem 19). Furthermore, we show that these functions vdp(t,z,ϵ) own with respect to ϵ a common asymptotic expansion v^(t,z,ϵ)=∑n≥0vn(t,z)ϵn which represents a formal series with bounded holomorphic coefficients on T×Hβ. We specify also the nature of this asymptotic expansion which turns out to be of Gevrey order (at most) 1/(χ+α)κ. Besides, since the aperture of the sectors Ep may be selected slightly larger than π/(χ+α)κ, the functions vdp can be identified as (χ+α)κ-sums of the formal series v^ (Theorem 21). By construction, the integer κ shows up in the highest order term of the operator P3 which is of irregular type of the form ϵΔDtδD(κ+1)+k0∂tδDRD(∂z), with k0=valt(P1), for some integers ΔD≥0, δD≥2 and a polynomial RD with complex coefficients. The rational number χ is built with the help of the integers ΔD, δD, k0, and κ and the rational numbers α, β; see (86). According to the fact that α, β are mainly related to constraints assumed on the polynomials P1 and P2 (see (66), (67)), we observe that the Gevrey order 1/(χ+α)κ of the asymptotic expansion involves information coming both from the highest irregular term and from the two polynomials P1and P2 that shape the turning point at t=0, whereas, in our previous contribution [3], the Gevrey order was exclusively stemming from the irregular singularity at t=0.The kind of equations with quadratic nonlinearity we investigate in this work is strongly related to singularly perturbed ODEs which are nonsingular at the origin of the formϵσdy/dt=F(t,y,a,ϵ) for some analytic functions F, small complex parameter ϵ, and a complex additional parameter a, described in the seminal joint paper by Canalis-Durand et al., see [6], where they study asymptotic properties of actual overstable solutions near a slow curve ϕ0(t) (meaning that F(t,ϕ0(t),a,0)≡0) in the case when the Jacobian ∂yF(t,ϕ0(t),a,0) is not invertible at t=0. The main notable difference is that we assume the origin to be at the same time a turning point and an irregular singularity. More precisely, with the rescaling map (t,ϵ)↦(T=ϵt,ϵ) the transformed equation (64) possesses a rational slow curve U0(T) and T=0 remains a turning point and an irregular singularity for this new equation.The construction of the distinguished solution performed in Section4 and the parametric Borel/Laplace summable character of these solutions shown in Section 7 are also intimately linked to recent developments of exact WKB analysis of formal and analytic solutions to second-order linear ODEs of Schrödinger type. Namely, let(5)ϵ2ψ′′t,ϵ=Qtψt,ϵbe a singularly perturbed ODE where ϵ is a small complex parameter and Q(t) is some polynomial with complex coefficients. WKB solutions of (5) are known as special solutions that are described as an exponential ψ^(t,ϵ)=exp(∫t0tS^s,ϵds) where the expression S^(t,ϵ) satisfies a so-calledRiccati equation(6)ϵ2S^′t,ϵ+ϵ2S^2t,ϵ=Qt.This last equation possesses formal power series solutions S^(t,ϵ)=S-1(t)/ϵ+∑n≥0Sn(t)ϵn, where S-1(t) satisfies the quadratic equation S-12(t)=Q(t). Once S-1(t)=±Q(t) is fixed, we get two formal solutions S^±(t,ϵ)=S-1(t)/ϵ+T^±(t,ϵ), where T^±(t,ϵ)∈C[[ϵ]] for any t∈U={t∈C∣Q(t)≠0}. Notice that T^±(t,ϵ) solves the first-order Riccati equation (7)ϵT^±′t,ϵ+2S-1tT^±t,ϵ+ϵT^±2t,ϵ+S-1′t=0with turning points at the roots of Q(t). Our main PDE (1) resembles this last one provided that S-1(t) is a polynomial and with the significant distinction that our equation only involves differential operators with irregular singularity at t=0. An essential feature of the theory is that the formal series T^±(t,ϵ) are 1-summable in suitable directions d∈R with respect to ϵ (that are related to the function ∫t0tS-1(s)ds) for any fixed t∈U. Different proofs of this fact can be found in [7–10]. Our second main statement, Theorem 21, can be considered as a similar contribution for some higher order PDEs of this latter result. Furthermore, in our study we are also able to describe the behaviour of our specific solutions near (t,ϵ)=(0,0).For more recent and advanced works related to WKB analysis and local/global studies of solutions to linear ODEs near turning points, we refer to contributions related to the 1D Schrödinger equation with simple poles [11], with merging pairs of simple poles and turning points [12], and with merging triplet of poles and turning points [13, 14] and for analytic continuation properties of the Borel transform (resurgence) of WKB expansions in the problem of confluence of two simple turning points we quote [15]. Concerning the structure of singular formal solutions to singularly perturbed linear systems of ODEs with turning points we point out [16] solving an old question of Wasow. We mention also preeminent studies on WKB analysis for higher order differential equations which reveal new Stokes phenomena giving rise to so-called virtual turning points [17, 18].In the framework of linear PDEs, normal forms for completely integrable systems near a degenerate point where two turning points coalesce have been obtained in [19], which is a first step toward the so-called Dubrovin conjecture which concerns the question of universal behaviour of generic solutions near gradient catastrophe of singularly Hamiltonian perturbations of first-order hyperbolic equations; see [20]. We mention also that sectorial analytic transformations to normal forms have been obtained for systems of singularly perturbed ODEs near a turning point with multiplicity using the recent approach of composite asymptotic expansions developed in [2]; see [21].The paper is organized as follows. In Section2, we recall the definition introduced in the work [3] of some weighted Banach spaces of continuous functions with exponential growth on unbounded sectors in C and with exponential decay on R. We analyze the continuity of specific multiplication and linear/nonlinear convolution operators acting on these spaces.In Section3, we remind the reader of basic statements concerning mk-Borel-Laplace transforms, a version of the classical Borel-Laplace maps already used in previous works [3, 22, 23] and Fourier transforms acting on exponentially flat functions.In Section4, we display our main problems and explain the leading strategy in order to solve them. It consists in four operations. In a first step, we restrict our inquiry for the sets of solutions to time rescaled function spaces; see (63). Then, we consider candidates for solutions to the resulting auxiliary problem (64) that are small perturbations of a so-called slow curve which solves a second-order algebraic equation and which may be singular at the origin in C. In a third step, we search again for time rescaled functions solutions for the associated problem (84) solved by the small perturbation of the slow curve; see (85). In the last step, we write down the convolution problem (95) solved by a suitable mκ-Borel transform of a formal solution to the attached problem (87).In Section5, we solve the main convolution problem (95) within the Banach spaces described in Section 2 using some fixed point theorem argument.In Section6, we provide a set of actual meromorphic solutions to our initial equation (61) by executing backwards the operations described in Section 4. In particular, we show that our singular functions actually solve problem (164) which is a factorized part of (61) with a more restrictive forcing term. Furthermore, the difference of any two neighboring solutions tends to 0 as ϵ tends to 0 faster than a function with exponential decay of order (χ+α)κ.In Section7, we show the existence of a common asymptotic expansion of Gevrey order 1/(χ+α)κ for the nonsingular parts of these solutions of (61) and (164) based on the flatness estimates obtained in Section 6 using a theorem by Ramis and Sibuya.


## 2. Banach Spaces with Exponential Growth and Exponential Decay

We denote byD(0,ρ) the open disc centered at 0 with radius ρ>0 in C and by D¯(0,ρ) its closure. Let Sd be an open unbounded sector in direction d∈R and E be an open sector with finite radius rE, both centered at 0 in C. By convention, these sectors do not contain the origin in C.We first give definitions of Banach spaces which already appear in our previous work [3].Definition 1.
Letβ>0 and μ>1 be real numbers. We denote by E(β,μ) the vector space of functions h:R→C such that (8)hmβ,μ=supm∈R⁡1+mμexpβmhmis finite. The space E(β,μ) endowed with the norm ·(β,μ) becomes a Banach space.As a direct consequence of Proposition  5 from [3], we notice the following.Proposition 2.
The Banach space(E(β,μ),·(β,μ)) is a Banach algebra for the convolution product (9)f⋆gm=∫-∞+∞fm-m1gm1dm1.Namely, there exists a constant C0>0 (depending on μ) such that (10)f⋆gmβ,μ≤C0fmβ,μgmβ,μfor all f,g∈E(β,μ).Definition 3.
Letν,ρ>0 and β>0, μ>1 be real numbers. Let κ≥1 and χ,α≥0 be integers. Let ϵ∈E. We denote by F(ν,β,μ,χ,α,κ,ϵ)d the vector space of continuous functions (τ,m)↦h(τ,m) on (D¯(0,ρ)∪Sd)×R, which are holomorphic with respect to τ on D(0,ρ)∪Sd and such that(11)hτ,mν,β,μ,χ,α,κ,ϵ=supτ∈D-0,ρ∪Sd,m∈R⁡1+mμexp⁡βm1+τ/ϵχ+α2κτ/ϵχ+αexp⁡-ντϵχ+ακhτ,mis finite. One can check that the normed space (Fν,β,μ,χ,α,κ,ϵd, ·ν,β,μ,χ,α,κ,ϵ) is a Banach space.Throughout the whole section, we keep the notations of Definitions1 and 3.In the next lemma, we check that some parameter depending functions with polynomial growth with respect to the variableτ and exponential decay with respect to the variable m, which will appear later on in our study (Section 5), belong to the Banach spaces described above.Lemma 4.
Letγ1≥0, γ2≥1 be integers. Let R~(X) be a polynomial that belongs to C[X] such that R~(im)≠0 for all m∈R. We take a function B~(m) located in E(β,μ) and we consider a continuous function aγ1,κ(τ,m) on (D¯(0,ρ)∪Sd)×R, holomorphic with respect to τ on D(0,ρ)∪Sd, such that (12)aγ1,κτ,m≤11+τκγ1R~imfor all τ∈D¯(0,ρ)∪Sd, all m∈R.
Then, the functionϵ-χγ2τγ2B~(m)aγ1,κ(τ,m) belongs to F(ν,β,μ,χ,α,κ,ϵ)d. Moreover, there exists a constant C1>0 (depending on κ and γ2) such that(13)ϵ-χγ2τγ2B~maγ1,κτ,mν,β,μ,χ,α,κ,ϵ≤C1B~mβ,μinfm∈R⁡R~imϵγ2αfor all ϵ∈E.Proof.
By definition of the norm and bearing in mind the constraint on the polynomialR~(X), we can write(14)ϵ-χγ2τγ2B~maγ1,κτ,mν,β,μ,χ,α,κ,ϵ≤supτ∈D¯0,ρ∪Sd,m∈R⁡1+mμexp⁡βmB~m×1+τ/ϵχ+α2κτ/ϵχ+αexp⁡-ντϵχ+ακϵ-χγ2τϵχ+αγ2ϵγ2χ+α11+τ/ϵχ+ακϵκχ+αγ1R~im≤B~mβ,μinfm∈RR~imϵγ2αsupx≥0⁡1+x2κxxγ2e-νxκ1+xκϵκχ+αγ1≤B~mβ,μinfm∈RR~imϵγ2αsupx≥0⁡1+x2κxxγ2e-νxκwhich yields the lemma since an exponential grows faster than any polynomial.The next proposition provides norm estimates for some linear convolution operators acting on the Banach spaces introduced above. These bounds are more accurate than the one supplied in Proposition  2 from [3]. These new estimates will be essential in Section 5 in order to solve problem (95). The improvements are due to the use of thorough upper bounds estimates of a generalized Mittag-Leffler function described in the proofs of Propositions  1 and  5 from [23].Proposition 5.
Letγj, 0≤j≤3, be real numbers with γ1≥0. Let R~(X) and R~D(X) be polynomials with complex coefficients such that deg(R~)≤deg(R~D) and with R~D(im)≠0 for all m∈R. We consider a continuous function aγ1,κ(τ,m) on (D¯(0,ρ)∪Sd)×R, holomorphic with respect to τ on D(0,ρ)∪Sd, such that (15)aγ1,κτ,m≤11+τκγ1R~Dimfor all τ∈D¯(0,ρ)∪Sd, all m∈R. We make the following assumptions:(16)1κ+γ3+1>0,γ2+γ3+2≥0,γ2>-1.
(1) If1+γ3≤0, then there exists a constant C2>0 (depending on ν,κ,γ2,γ3 and R~(X),R~D(X)) such that(17)ϵ-γ0aγ1,κτ,mR~imτκ∫0τκτκ-sγ2sγ3fs1/κ,mdsν,β,μ,χ,α,κ,ϵ≤C2ϵχ+ακγ2+γ3+2-γ0fτ,mν,β,μ,χ,α,κ,ϵfor all f(τ,m)∈F(ν,β,μ,χ,α,κ,ϵ)d.
(2) If1+γ3>0 and γ1≥1+γ3, then there exists a constant C2′>0 (depending on ν, κ, γ1, γ2, γ3 and R~(X), R~D(X)) such that(18)ϵ-γ0aγ1,κτ,mR~imτκ∫0τκτκ-sγ2sγ3fs1/κ,mdsν,β,μ,χ,α,κ,ϵ≤C2′ϵχ+ακγ2+γ3+2-γ0-χ+ακγ1fτ,mν,β,μ,χ,α,κ,ϵfor all f(τ,m)∈F(ν,β,μ,χ,α,κ,ϵ)d.Proof.
By definition of the norm, we can write(19)A=ϵ-γ0aγ1,κτ,mR~imτκ∫0τκτκ-sγ2sγ3fs1/κ,mdsν,β,μ,χ,α,κ,ϵ≤supτ∈D¯0,ρ∪Sd,m∈R⁡1+mμexp⁡βm1+τ/ϵχ+α2κτ/ϵχ+αexp⁡-ντϵχ+ακϵ-γ011+τκγ1R~imR~Dim×τκ∫0τκ1+mμexpβm1+s2/ϵχ+α2κs1/κ/ϵχ+αexp⁡-νsϵχ+ακfs1/κ,mAτ,s,m,ϵds,where (20)Aτ,s,m,ϵ=11+mμexpβmexp⁡νs/ϵχ+ακ1+s2/ϵχ+α2κs1/κϵχ+ατκ-sγ2sγ3.Again by the definition of the norm of f and by the constraints on the polynomials R, RD, we deduce that(21)A≤C2.1ϵsupm∈RR~imR~Dimfτ,mν,β,μ,χ,α,κ,ϵ,where(22)C2.1ϵ=supτ∈D¯0,ρ∪Sd⁡1+τ/ϵχ+α2κτ/ϵχ+αexp⁡-ντϵχ+ακϵ-γ011+τκγ1×τκ∫0τκexp⁡νh/ϵχ+ακ1+h2/ϵχ+α2κh1/κϵχ+ατκ-hγ2hγ3dh.We perform the change of variable h=ϵχ+ακh′ inside the integral which is a part of C2.1(ϵ) that yields(23)C2.1ϵ=supτ∈D¯0,ρ∪Sd⁡1+τ/ϵχ+α2κτ/ϵχ+αexp⁡-ντϵχ+ακϵ-γ011+τκγ1×τκ∫0τκ/ϵχ+ακeνh′1+h′2h′1/κτκϵχ+ακ-h′γ2h′γ3dh′ϵχ+ακγ2+γ3+1.As a result, we obtain the bounds(24)C2.1ϵ≤ϵχ+ακγ2+γ3+1-γ0+χ+ακsupx≥0⁡1+x2x1/κe-νxx1+ϵχ+ακxγ1Gx,where(25)Gx=∫0xeνh′1+h′2h′1/κ+γ3x-h′γ2dh′.We now proceed as in Proposition  1 of [23]. We split the function G(x) into two pieces and study them separately. Namely, we decompose G(x)=G1(x)+G2(x), where (26)G1x=∫0x/2eνh′1+h′2h′1/κ+γ3x-h′γ2dh′,G2x=∫x/2xeνh′1+h′2h′1/κ+γ3x-h′γ2dh′.We first provide estimates for G1(x).
(a) Assume that-1<γ2<0. We see that (x-h′)γ2≤(x/2)γ2 for all 0≤h′≤x/2, for x>0. Hence, from the first constraint of (16), we get (27)G1x≤x2γ2eνx/2∫0x/2h′1/κ+γ3dh′=x2γ2eνx/2x/21/κ+γ3+11/κ+γ3+1for all x≥0. Subsequently, we obtain(28)supx≥0⁡1+x2x1/κe-νxx1+ϵχ+ακxγ1G1x≤supx≥0⁡1+x2x1/κe-νxxG1xwhich is finite due to the second assumption of (16).
(b) Assume thatγ2>0. We notice that (x-h′)γ2≤xγ2 for all 0≤h′≤x/2, for x>0. Therefore, again from the first constraint of (16) we get (29)G1x≤xγ2eνx/2∫0x/2h′1/κ+γ3dh′=xγ2eνx/2x/21/κ+γ3+11/κ+γ3+1for all x≥0. Consequently, we obtain(30)supx≥0⁡1+x2x1/κe-νxx1+ϵχ+ακxγ1G1x≤supx≥0⁡1+x2x1/κe-νxxG1xwhich is finite due to the second assumption of (16).
In a second step, we studyG2(x).
We see that1+(h′)2≥1+(x/2)2 for all x/2≤h′≤x. Hence,(31)G2x≤11+x/22∫x/2xeνh′h′1/κ+γ3x-h′γ2dh′≤11+x/22G2.1x,where (32)G2.1x=∫0xeνh′h′1/κ+γ3x-h′γ2dh′for all x≥0. Taking account of the estimates (18) in [23] which are deduced from the asymptotic behaviour for large x of the generalized Mittag-Leffler function Eα,β(x)=∑n≥0xn/Γ(β+nα), for α,β>0, we get a constant K2.1>0 (that depends on ν, κ, γ2, γ3) such that(33)G2.1x≤K2.1x1/κ+γ3eνxfor all x≥1, provided the first and last constraints of (16) hold.
(1) We consider the first case when1+γ3≤0.
Bearing in mind (31) and (33), we deduce that(34)supx≥1⁡1+x2x1/κe-νxx1+ϵχ+ακxγ1G2x≤supx≥1⁡1+x21+x/22K2.1x1+γ3which is finite. On the other hand, when 0≤x<1, we make the change of variable h′=xu′ inside G2.1x and, taking (31) into account, we get(35)sup0≤x<1⁡1+x2x1/κe-νxx1+ϵχ+ακxγ1G2x≤sup0≤x<1⁡1+x21+x/22e-νxxx1/κx1/κ+γ3+γ2+1×∫01eνxu′u′1/κ+γ31-u′γ2du′which is finite provided that the constraints (16) are fulfilled.
(2) We examine the second case when1+γ3>0 and γ1≥γ3+1.
We use this time the fact that1+ϵ(χ+α)κx≥ϵ(χ+α)κx for all x≥1 and the bounds (33) in order to get(36)supx≥1⁡1+x2x1/κe-νxx1+ϵχ+ακxγ1G2x≤ϵ-χ+ακγ1K2.1supx≥1⁡1+x21+x/22xγ3+1xγ1.On the other hand, the bounds on the domain 0≤x<1 have already been treated above owing to (35).
Finally, gathering (21), (24), (28), (30), (34), (35), and (36) yields the statement of Proposition 5.The forthcoming proposition presents norm estimates for some bilinear convolution operators acting on the aforementioned Banach spaces.Proposition 6.
There exists a constantC3>0 (depending on μ and κ) such that(37)τκ-1∫0τκ∫-∞+∞fτκ-s′1/κ,m-m1gs′1/κ,m11τκ-s′s′ds′dm1ν,β,μ,χ,α,κ,ϵ≤C3ϵχ+αfτ,mν,β,μ,χ,α,κ,ϵgτ,mν,β,μ,χ,α,κ,ϵfor all f(τ),g(τ)∈F(ν,β,μ,χ,α,κ,ϵ)d.Proof.
We follow the same guidelines as in the proof of Proposition  3 from [3]. By definition of the norm, we can write(38)B=τκ-1∫0τκ∫-∞+∞fτκ-s′1/κ,m-m1gs′1/κ,m11τκ-s′s′ds′dm1ν,β,μ,χ,α,κ,ϵ=supτ∈D¯0,ρ∪Sd,m∈R1+mμexp⁡βm1+τ/ϵχ+α2κτ/ϵχ+αexp⁡-ντϵχ+ακ×τκ-1∫0τκ∫-∞+∞1+m-m1μexp⁡βm-m1×1+τκ-s′2/ϵχ+α2κτκ-s′1/κ/ϵχ+αexp⁡-ντκ-s′ϵχ+ακfτκ-s′1/κ,m-m1×1+m1μexpβm11+s′2/ϵχ+α2κs′1/κ/ϵχ+αexp⁡-νs′ϵχ+ακgs′1/κ,m1×Bτ,s,m,m1ds′dm1,where(39)Bτ,s,m,m1=exp⁡-βm-m1exp⁡-βm11+m-m1μ1+m1μs′1/κτκ-s′1/κ/ϵ2χ+α1+τκ-s′2/ϵχ+α2κ1+s′2/ϵχ+α2κ×exp⁡ντκ-s′ϵχ+ακexp⁡νs′ϵχ+ακ1τκ-s′s′.By definition of the norms of f and g and according to the triangular inequality m≤m-m1+m1 for all m,m1∈R, we deduce that(40)B≤C3ϵfτ,mν,β,μ,χ,α,κ,ϵgτ,mν,β,μ,χ,α,κ,ϵ,where(41)C3ϵ=supτ∈D¯0,ρ∪Sd,m∈R⁡1+mμ1+τ/ϵχ+α2κτ/ϵχ+αexp⁡-ντϵχ+ακτκ-1×∫0τκ∫-∞+∞11+m-m1μ1+m1μh′1/κτκ-h′1/κϵ2χ+α11+τκ-h′2/ϵχ+α2κ1+h′2/ϵχ+α2κ×expντκ-h′ϵχ+ακexp⁡νh′ϵχ+ακ1τκ-h′h′dh′dm1.We provide upper bounds that can be split in two parts,(42)C3ϵ≤C3.1C3.2ϵ,where(43)C3.1=supm∈R⁡1+mμ∫-∞+∞11+m-m1μ1+m1μdm1is finite under the condition that μ>1 according to Lemma  4 of [24] and(44)C3.2ϵ=supτ∈D¯0,ρ∪Sd⁡1+τ/ϵχ+α2κτ/ϵχ+ατκ-1×∫0τκh′1/κτκ-h′1/κ/ϵ2χ+α1+τκ-h′2/ϵχ+α2κ1+h′2/ϵχ+α2κ1τκ-h′h′dh′.We carry out the change of variable h′=ϵ(χ+α)κh inside the integral piece of C3.2(ϵ) which yields the bounds(45)C3.2ϵ=supτ∈D¯0,ρ∪Sd⁡1+τ/ϵχ+α2κτ/ϵχ+ατκ-1×∫0τκ/ϵχ+ακh1/κτκ/ϵχ+ακ-h1/κ1+τκ/ϵχ+ακ-h21+h21τκ/ϵχ+ακ-hh1ϵχ+ακdh≤1ϵχ+αsupx≥0⁡Bx,where (46)Bx=1+x2x2/κx∫0xh1/κx-h1/κ1+x-h21+h21x-hhdh.A change of variable h=xu in this last expression followed by a partial fraction decomposition allow us to write(47)Bx=1+x2∫0111+x21-u21+x2u211-u1-1/κu1-1/κdu=1+x2x2+4∫013-2u1+x21-u211-u1-1/κu1-1/κdu+1+x2x2+4∫012u+11+x2u211-u1-1/κu1-1/κduwhich acquaints us with the fact that B(x) is finite provided that κ≥1 and bounded on R+ with respect to x.
At last, collecting (38), (40), (42), (43), (45), and (47) leads to the statement of Proposition 6.


## 3. Borel-Laplace and Fourier Transforms

In this section, we review some basic statements concerning ak-Borel summability method of formal power series which is a slightly modified version of the more classical procedure (see [5], Section  3.2). This novel version has already been used in works such as [3, 22] when studying Cauchy problems under the presence of a small perturbation parameter. We remind also the reader of the definition of Fourier inverse transform acting on functions with exponential decay.Definition 7.
Letk≥1 be an integer. Let (mk(n))n≥1 be the sequence (48)mkn=Γnk=∫0∞tn/k-1e-tdt,n≥1.Let (E,·E) be a complex Banach space. We say a formal power series (49)X^T=∑n=1∞anTn∈TETis mk-summable with respect to T in the direction d∈0,2π if the following assertions hold: (1)
There existsρ>0 such that the mk-Borel transform of X^, Bmk(X^), is absolutely convergent for τ<ρ, where (50)BmkX^τ=∑n=1∞anΓn/kτn∈τEτ.(2)
The seriesBmk(X^) can be analytically continued in a sector S={τ∈C⋆:d-argτ<δ} for some δ>0. In addition to this, the extension is of exponential growth at most k in S, meaning that there exist C,K>0 such that (51)BmkX^τE≤CeKτk,τ∈S.Under these assumptions, the vector valued Laplace transform of Bmk(X^) along direction d is defined by (52)LmkdBmkX^T=k∫LγBmkX^ue-u/Tkduu,where Lγ is the path parametrized by u∈[0,∞)↦ueiγ, for some appropriate direction γ depending on T, such that Lγ⊆S and cos(k(γ-arg(T)))≥Δ>0 for some Δ>0.
The functionLmkd(Bmk(X^)) is well defined and turns out to be a holomorphic and bounded function in any sector of the form Sd,θ,R1/k={T∈C⋆:T<R1/k,d-argT<θ/2}, for some π/k<θ<π/k+2δ and 0<R<Δ/K. This function is known as the mk-sum of the formal power series X^(T) in the direction d.The following are some elementary properties concerning themk-sums of formal power series which will be crucial in our procedure.(1) The functionLmkd(Bmk(X^))(T) admits X^(T) as its Gevrey asymptotic expansion of order 1/k with respect to T in Sd,θ,R1/k. More precisely, for every π/k<θ1<θ, there exist C,M>0 such that (53)LmkdBmkX^T-∑p=1n-1apTpE≤CMnΓ1+nkTn,for every n≥2 and T∈Sd,θ1,R1/k. Watson’s lemma (see Proposition  11 p. 75 in [25]) allows us to affirm that LmkdBmkX^T is unique provided that the opening θ1 is larger than π/k.(2) WheneverE is a Banach algebra, the set of holomorphic functions having Gevrey asymptotic expansion of order 1/k on a sector with values in E turns out to be a differential algebra (see Theorems  18, 19, and 20 in [25]). This and the uniqueness provided by Watson’s lemma allow us to obtain some properties on mk-summable formal power series in direction d.By⋆ we denote the product in the Banach algebra and also the Cauchy product of formal power series with coefficients in E. Let X^1,X^2∈TE[[T]] be mk-summable formal power series in direction d. Let q1≥q2≥1 be integers. Then X^1+X^2, X^1⋆X^2, and Tq1∂Tq2X^1, which are elements of TE[[T]], are mk-summable in direction d. Moreover, one has (54)LmkdBmkX^1T+LmkdBmkX^2T=LmkdBmkX^1+X^2T,LmkdBmkX^1T⋆LmkdBmkX^2T=LmkdBmkX^1⋆X^2T,Tq1∂Tq2LmkdBmkX^1T=LmkdBmkTq1∂Tq2X^1T,for every T∈Sd,θ,R1/k.The next proposition is written without proof for it can be found in [3], Proposition  6.Proposition 8.
Letf^(t)=∑n≥1fntn and g^(t)=∑n≥1gntn which belong to Et, where (E,·E) is a Banach algebra. Let k,m≥1 be integers. The following formal identities hold. (55)Bmktk+1∂tf^tτ=kτkBmkf^tτ,Bmktmf^tτ=τkΓm/k∫0τkτk-sm/k-1Bmkf^ts1/kdss,Bmkf^t⋆g^tτ=τk∫0τkBmkf^tτk-s1/k⋆Bmkg^ts1/k1τk-ssds.In the last part of the section, we recall without proofs some properties of the inverse Fourier transform acting on continuous functions with exponential decay onR; see [3], Proposition  7 for more details.Proposition 9.
(1) Letf:R→R be a continuous function with a constant C>0 such that fm≤Cexp(-βm) for all m∈R, for some β>0. The inverse Fourier transform of f is defined by the integral representation (56)F-1fx=12π1/2∫-∞+∞fmexp⁡ixmdmfor all x∈R. It turns out that the function F-1(f) extends to an analytic function on the horizontal strip(57)Hβ=z∈C∣Imz<β.Let ϕm=imf(m). Then, we have the commuting relation(58)∂zF-1fz=F-1ϕzfor all z∈Hβ.
(2) Letf,g∈E(β,μ) and let ψ(m)=(1/(2π)1/2)f⋆g(m), the convolution product of f and g, for all m∈R. From Proposition 2, we know that ψ∈E(β,μ). Moreover, the following formula(59)F-1fzF-1gz=F-1ψzholds for all z∈Hβ.


## 4. Layout of the Main Nonlinear PDE and Related Auxiliary Problems

Letq,M,p≥0, D≥2 be integers. For all 0≤l≤q, let kl, ml be nonnegative integers and al be complex numbers with a0≠0 such that kl<kl+1 for l∈{0,…,q-1}. For all 0≤l≤M, we consider nonnegative integers hl, μl and complex numbers cl with c0≠0 such that hl<hl+1 for l∈{0,…,M-1}. For all 0≤l≤p, we denote by nl and bl nonnegative integers such that bl<bl+1 for l∈{0,…,p-1}. For 1≤l≤D, we set nonnegative integers Δl, dl, and δl such that 1≤δl<δl+1 for l∈{1,…,D-1}.LetQ(X),Rl(X)∈C[X], 1≤l≤D, be polynomials which can be factorized as Q(X)=XvQ~(X), Rl(X)=XvR~l(X), for some common integer v≥1, where Q~(X) and R~l(X) are polynomials that satisfy(60)degQ~=degR~D≥degR~l,Q~im≠0,R~Dim≠0for all m∈R and all 1≤l≤D-1.We consider the following nonlinear singularly perturbed PDE:(61)Q∂z∑l=0qalϵmltklut,z,ϵ+∑l=0Mclϵμlthlu2t,z,ϵ=∑j=0pbjzϵnjtbj+∑l=1DϵΔltdl∂tδlRl∂zut,z,ϵ.The coefficients bj(z) are constructed as follows. For all 0≤j≤p, we consider functions m↦B~j(m) that belong to the Banach space E(β,μ) for some μ>1 and β>0. We define Bj(m)=(im)vB~j(m) where v is the integer introduced above, for 0≤j≤p. We set(62)bjz=F-1m⟼Bjmz,0≤j≤p,where F-1 denotes the Fourier inverse transform defined in Proposition 9. From (58), it turns out by construction that one can write bj(z)=∂zvb~j(z), where b~j(z) is the inverse Fourier transform of B~j(m).Remark 10.
The reason why we make these factorizations hypotheses on the polynomialsQ(X), Rl(X), and the functions Bj(m) will be explained later on in Remark 14 of next section and is related to the construction of the Banach spaces in Section 2 and their Fourier inverse transforms.
Within this work, we will search for time rescaled solutions of (61) of the form(63)ut,z,ϵ=ϵβUϵαt,z,ϵ,where α,β∈Q are two rational numbers and α>0. Then, the expression U(T,z,ϵ) needs to formally solve the following nonlinear PDE:(64)Q∂z∑l=0qalϵml+β-αklTklUT,z,ϵ+∑l=0Mclϵμl+2β-αhlThlU2T,z,ϵ=∑j=0pbjzϵnj-αbjTbj+∑l=1DϵΔl+αδl-dl+βTdlRl∂z∂TδlUT,z,ϵ.


### 4.1. Construction of a Distinguished Solution

We make the additional assumption thatα, β set above can be chosen in such a way that the following inequalities(65)Δl+αδl-dl+β>0,nj-αbj>0for all 1≤l≤D, 0≤j≤p and(66)ml+β-αkl=0,mj+β-αkj>0for all 0≤l≤s and all s+1≤j≤q, for some integer 0≤s≤q-1, together with(67)μl+2β-αhl=0,μj+2β-αhj>0for all 0≤l≤s′ and all s′+1≤j≤M, for some integer 0≤s′≤M-1, hold.Remark 11.
In the caseq=1, k0,k1≥1, the roots of the polynomial (in t) P(t,ϵ)=a0ϵm0tk0+a1ϵm1tk1 all have modulus equal to (68)a1a01/k0-k1ϵm1-m0/k0-k1except the trivial root 0. The constraints (66) imply in particular that m1-m0>α(k1-k0). As a result, all the nonvanishing roots of P(t,ϵ) tend to ∞ as ϵ tends to 0 and 0 is therefore the only root (with order k0) of P(t,ϵ) in the vicinity of 0 as ϵ stays near the origin.
Let us assume that the expressionU(T,z,ϵ) is allowed to be written as a perturbation series with respect to ϵ:(69)UT,z,ϵ=U0T+∑n≥1UnT,zϵn,where the constant term U0(T) is taken independently of z and is not identically equal to 0. The coefficient U0(T) is called theslow curve of (64) in the terminology of [6].
In the following, we make the assumption thatU0(T) solves the following second-order algebraic equation:(70)∑l=0salTklU0T+∑l=0s′clThlU0T2=0.As U0(T) is not identically vanishing, it must be equal to -(∑l=0salTkl)/(∑l=0s′clThl). Bearing in mind that a0,c0≠0, we get its asymptotic behaviour(71)U0T~-a0c0Tk0-h0as T tends to 0.Remark 12.
Under the hypotheses (60) and (62), we observe, by factoring out the operator ∂zv from (64), that U(T,z,ϵ) must solve the related PDE(72)Q~∂z∑l=0qalϵml+β-αklTklUT,z,ϵ+∑l=0Mclϵμl+2β-αhlThlU2T,z,ϵ=∑j=0pb~jzϵnj-αbjTbj+FT,z,ϵ+∑l=1DϵΔl+αδl-dl+βTdlR~l∂z∂TδlUT,z,ϵ,where the forcing term F(T,z,ϵ) is a polynomial in z of degree less than v-1. According to the assumptions (65), (66), and (67) and using the fact that Q~(0)≠0, by taking ϵ=0 into (72), we see that the constraint (70) is equivalent to the fact that F(T,z,0)≡0. The precise shape of the term F(T,z,ϵ) will be given later in Section 6; see (183).
In afirst step, we express U(T,z,ϵ) as a small perturbation of U0(T) that can be expressed in the form (73)U0T=-a0c0Tk0-h0-a0c0Tk0-h0JT,where J(T)=∑j≥1JjTj is a convergent series near T=0; namely,(74)UT,z,ϵ=-a0c0Tk0-h0-a0c0Tk0-h0JT+TγVT,z,ϵfor some integer γ∈Z and some expression V(T,x,ϵ). By plugging this last expansion inside (64) and using the Leibniz rule, we get(75)Q∂z∑l=0salTkl+∑l=s+1qalϵml+β-αklTkl-a0c0Tk0-h0-a0c0Tk0-h0JT+TγVT,z,ϵ+∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl-a0c0Tk0-h0-a0c0Tk0-h0JT+TγVT,z,ϵ2=∑j=0pbjzϵnj-αbjTbj+∑l=1DϵΔl+αδl-dl+βTdlRl∂z-a0c0∏d=0δl-1k0-h0-dTk0-h0-δl-a0c0∑q1+q2=δlδl!q1!q2!∂Tq1Tk0-h0∂Tq2JT+∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dTγ-q1∂Tq2VT,z,ϵ,where we put ∏d=0-1(γ-d)=1 by convention. At this level, we observe the important fact that the coefficient in front of Q(∂z)V(T,z,ϵ) contains the term a0Tk0+γ-2(a0/c0)Tk0-h0+γc0Th0=-a0Tk0+γ that we want to set apart. As a result, we get the following equation satisfied by V(T,z,ϵ):(76)Q∂zVT,z,ϵ-a0Tk0+γ+∑l=1salTkl+∑l=s+1qalϵml+β-αklTklTγ-2a0c0Tk0-h0+γ∑l=1s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl-2a0c0Tk0-h0+γJT×∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl+Q∂zV2T,z,ϵT2γ∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl=∑j=0pbjzϵnj-αbjTbj+∑l=1DϵΔl+αδl-dl+β∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dTdl+γ-q1Rl∂z∂Tq2VT,z,ϵ.We now introduce some additional constraints on the integers γ, kl, hj, bh, for 0≤l≤q, 0≤j≤M, 0≤h≤p and dl, δl, for 1≤l≤D. Namely, we impose that the following inequalities hold:(77)k0-hp≤γfor 0≤p≤M, together with(78)γ≤bj-k0for all 0≤j≤p and finally(79)k0<dD-δD,k0≤dl-δlfor all 1≤l≤D-1.Remark 13.
(1) For the casek0>h0, from (77), we need γ≥k0-h0>0. As a consequence of (78), we get that bj>k0, for 0≤j≤p. Let, for instance, q=1, M=1, p=0, and D=2. We set α=2, β=1, γ=6, κ=1 and we choose the powers of t and ϵ in the coefficients of (61) as follows:(80)m0=3,k0=2,m1=6,k1=3,μ0=0,h0=1,μ1=3,h1=2,n0=19,b0=9,Δ1=12,d1=5,δ1=1,Δ2=20,d2=6,δ2=2.For these data, we can check that the constraints (65), (66), (67), (77), (78), and (79) above are fulfilled. Moreover, all the forthcoming requirements (88), (105), (106), (107), and (184) stated in Theorem 19 are also verified. In this special case, the main equation (61) writes(81)Q∂za0ϵ3t2+a1ϵ6t3ut,z,ϵ+c0t+c1ϵ3t2u2t,z,ϵ=b0zϵ19t9+ϵ12t5∂tR1∂zut,z,ϵ+ϵ20t6∂t2R2∂zut,z,ϵ.We can divide this last equation by t, but not by ϵ, and the resulting equation still possesses a turning point and an irregular singularity at t=0.
(2) For the caseh0>k0, we may take γ<0 and hence one can choose some bj<k0 for some 0≤j≤p. Let, for instance, q=1, M=1, p=0, and D=2. We choose α=2, β=-1, γ=-2, and κ=1 and we select the powers of t and ϵ in the coefficients of (61) as follows:(82)m0=5,k0=2,m1=10,k1=4,μ0=14,h0=6,μ1=19,h1=8,n0=3,b0=1,Δ1=10,d1=5,δ1=1,Δ2=12,d2=6,δ2=2.For these data, we can figure out that the constraints (65), (66), (67), (77), (78), and (79) above are satisfied. Moreover, all the forthcoming requirements (88), (105), (106), (107), and (184) stated in Theorem 19 are also verified. In this particular case, the main equation (61) writes(83)Q∂za0ϵ5t2+a1ϵ10t4ut,z,ϵ+c0ϵ14t6+c1ϵ19t8u2t,z,ϵ=b0zϵ3t+ϵ10t5∂tR1∂zut,z,ϵ+ϵ12t6∂t2R2∂zut,z,ϵ.We can divide this latter equation by ϵ3 and by t. The corresponding equation still suffers the presence of a turning point and an irregular singularity at t=0.
In asecond step, we divide the left- and right-hand sides of (76) by the monomial Tk0+γ. We obtain the following equation:(84)Q∂zVT,z,ϵ-a0+∑l=1salTkl+∑l=s+1qalϵml+β-αklTklT-k0-2a0c0T-h0∑l=1s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl-2a0c0T-h0JT×∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl+Q∂zV2T,z,ϵT-k0+γ∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl=∑j=0pbjzϵnj-αbjTbj-k0-γ+∑l=1DϵΔl+αδl-dl+β∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dTdl-k0-q1Rl∂z∂Tq2VT,z,ϵ.Notice that the additional constraints (65), (66), (67) and (77), (78), (79) ensure that the coefficients of the PDE (84) are analytic with respect to T and ϵ on a neighborhood of the origin in C2. Moreover, the coefficient of Q(∂z)V(T,z,ϵ) is invertible at T=0 since a0≠0. We will see later that this fact is essential in order to solve this equation within some function space of analytic functions.
We look for solutions which are rescaled in time of the form(85)VT,z,ϵ=VϵχT,z,ϵ,where(86)χ=ΔD+αδD-dD+βdD-k0-δD.As a result, provided that T=ϵχT holds, the expression V(T,z,ϵ) is supposed to solve the following equation:(87)Q∂zVT,z,ϵ-a0+∑l=1salϵ-χkl-k0Tkl-k0+∑l=s+1qalϵml+β-αkl-χkl-k0Tkl-k0-2a0c0∑l=1s′clϵ-χhl-h0Thl-h0+∑l=s′+1Mclϵμl+2β-αhl-χhl-h0Thl-h0-2a0c0Jϵ-χT×∑l=0s′clϵ-χhl-h0Thl-h0+∑l=s′+1Mclϵμl+2β-αhl-χhl-h0Thl-h0+Q∂zV2T,z,ϵϵ-χ-k0+γT-k0+γ∑l=0s′clϵ-χhlThl+∑l=s′+1Mclϵμl+2β-αhl-χhlThl=∑j=0pbjzϵnj-αbj-χbj-k0-γTbj-k0-γ+∑l=1D-1ϵΔl+αδl-dl+β×∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dϵ-χdl-k0-q1Tdl-k0-q1Rl∂zϵχq2∂Tq2VT,z,ϵ+ϵΔD+αδD-dD+β∑q1+q2=δD,q1≥1δD!q1!q2!∏d=0q1-1γ-dϵ-χdD-k0-q1TdD-k0-q1×RD∂zϵχq2∂Tq2VT,z,ϵ+TdD-k0RD∂z∂TδDVT,z,ϵ.
We make further assumptions on the coefficientsdl and δl for 1≤l≤D which are stronger than the constraint (79). Assume the existence of integers κ≥1 and dl,0≥1 such that(88)dD-k0=δDκ+1,dl-k0=δlκ+1+dl,0for all 1≤l≤D-1. Then, for all 1≤l≤D, and all integers q1≥0, q2≥0 with q1+q2=δl we deduce the existence of a nonnegative integer dl,q1,q2 which is larger than 1 except dD,0,δD=0 such that(89)dl-k0-q1=κ+1q2+dl,q1,q2.Indeed, if one puts dD,0=0, from (88), we can write(90)dl,q1,q2=dl-k0-q1-κ+1q2=δlκ+1+dl,0-q1-κ+1q2=q1+q2κ+1+dl,0-q1-κ+1q2=q1κ+dl,0.According to (88) and (89), with the help of formula (8.7) from [26], p. 3630, we can expand the following pieces appearing in (87) satisfied by V(T,z,ϵ):(91)TdD-k0∂TδDVT,z,ϵ=Tκ+1∂TδD+∑1≤p≤δD-1AδD,pTκδD-pTκ+1∂TpVT,z,ϵ,Tdl-k0-δl-1∂TVT,z,ϵ=Tdl,δl-1,1Tκ+1∂TVT,z,ϵ,Tdl-k0-q1∂Tq2VT,z,ϵ=Tdl,q1,q2Tκ+1q2∂Tq2VT,z,ϵ=Tdl,q1,q2Tκ+1∂Tq2+∑1≤p≤q2-1Aq2,pTκq2-pTκ+1∂TpVT,z,ϵfor all 1≤l≤D and all integers q1≥0 and q2≥2 such that q1+q2=δl, for some real constants AδD,p, 1≤p≤δD-1, and Aq2,p, 1≤p≤q2-1.
In athird step, let us assume that the expression V(T,z,ϵ) has a formal power series expansion(92)VT,z,ϵ=∑n≥1Vnz,ϵTn,where each coefficient Vn(z,ϵ) is defined as an inverse Fourier transform (93)Vnz,ϵ=F-1m⟼ωnm,ϵzfor some function m↦ωn(m,ϵ) belonging to the Banach space E(β,μ) and depending holomorphically on ϵ on some punctured disc D(0,ϵ0)∖{0} centered at 0 with radius ϵ0>0. We consider the formal power series (94)ωκτ,m,ϵ=∑n≥1ωnm,ϵΓn/κτnobtained by formally applying mκ-Borel transform with respect to T and Fourier transform with respect to z to the power series (92). The constraints (88) are introduced in such a way that ωκ(τ,m,ϵ) satisfies some integral equation by making use of the properties of the mκ-Borel transform of formal series and Fourier inverse transforms described in Propositions 8 and 9 with the help of the prepared expansions (91). Namely, after division by the power (im)v, which is by construction a common factor of the functions Q(im), Rl(im), and Bj(m) for 1≤l≤D, 0≤j≤p, we get the new problem(95)Lτ,m,ϵωκτ,m,ϵ=Rτ,m,ϵωκτ,m,ϵwith vanishing initial data ωκ(0,m,ϵ)≡0, where(96)Lτ,m,ϵωκτ,m,ϵ=Q~im-a0ωκτ,m,ϵ+∑l=1salϵ-χkl-k0τκΓkl-k0/κ∫0τκτκ-skl-k0/κ-1ωκs1/κ,m,ϵdss+∑l=s+1qalϵml+β-αkl-χkl-k0τκΓkl-k0/κ∫0τκτκ-skl-k0/κ-1ωκs1/κ,m,ϵdss-2a0c0∑l=1s′clϵ-χhl-h0τκΓhl-h0/κ∫0τκτκ-shl-h0/κ-1ωκs1/κ,m,ϵdss+∑l=s′+1Mclϵμl+2β-αhlϵ-χhl-h0τκΓhl-h0/κ∫0τκτκ-shl-h0/κ-1ωκs1/κ,m,ϵdss-2a0c0∑l=0s′cl∑j≥1Jjϵ-χhl-h0+jτκΓhl-h0+j/κ∫0τκτκ-shl-h0+j/κ-1ωκs1/κ,m,ϵdss+∑l=s′+1Mcl∑j≥1Jjϵμl+2β-αhlϵ-χhl-h0+jτκΓhl-h0+j/κ∫0τκτκ-shl-h0+j/κ-1ωκs1/κ,m,ϵdss+Q~im∑l=0s′clϵ-χ-k0+γ+hlτκΓ-k0+γ+hl/κ∫0τκτκ-s-k0+γ+hl/κ-1×s∫0s∫-∞+∞12π1/2ωκs-s′1/κ,m-m1,ϵωκs′1/κ,m1,ϵ1s-s′s′ds′dm1dss+∑l=s′+1Mclϵμl+2β-αhl-χhl-k0+γτκΓ-k0+γ+hl/κ∫0τκτκ-s-k0+γ+hl/κ-1×s∫0s∫-∞+∞12π1/2ωκs-s′1/κ,m-m1,ϵωκs′1/κ,m1,ϵ1s-s′s′ds′dm1dss,(97)Rτ,m,ϵωκτ,m,ϵ=∑j=0pB~jmϵnj-αbj-χbj-k0-γτbj-k0-γΓbj-k0-γ/κ+∑l=1D-1ϵΔl+αδl-dl+β×∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dϵ-χdl-k0-q1-q2R~lim×τκΓdl,q1,q2/κ∫0τκτκ-sdl,q1,q2/κ-1κq2sq2ωκs1/κ,m,ϵdss+∑1≤p≤q2-1Aq2,pτκΓdl,q1,q2+κq2-p/κ∫0τκτκ-sdl,q1,q2+κq2-p/κ-1κpspωκs1/κ,m,ϵdss+ϵΔD+αδD-dD+β×∑q1+q2=δD,q1≥1δD!q1!q2!∏d=0q1-1γ-dϵ-χdD-k0-q1-q2R~Dim×τκΓdD,q1,q2/κ∫0τκτκ-sdD,q1,q2/κ-1κq2sq2ωκs1/κ,m,ϵdss+∑1≤p≤q2-1Aq2,pτκΓdD,q1,q2+κq2-p/κ∫0τκτκ-sdD,q1,q2+κq2-p/κ-1κpspωκs1/κ,m,ϵdss+R~DimκτκδDωκτ,m,ϵ+∑1≤p≤δD-1AδD,pτκΓκδD-p/κ×∫0τκτκ-sκδD-p/κ-1κpspωκs1/κ,m,ϵdss.By convention, the two sums ∑1≤p≤q2-1[⋯] appearing in (97) are vanishing provided that q2∈{0,1}.Remark 14.
The hypotheses (60) and (62) ensure that (87) does not contain terms that involve isolated polynomials in T which are not inverse Fourier transformable.


## 4.1. Construction of a Distinguished Solution

We make the additional assumption thatα, β set above can be chosen in such a way that the following inequalities(65)Δl+αδl-dl+β>0,nj-αbj>0for all 1≤l≤D, 0≤j≤p and(66)ml+β-αkl=0,mj+β-αkj>0for all 0≤l≤s and all s+1≤j≤q, for some integer 0≤s≤q-1, together with(67)μl+2β-αhl=0,μj+2β-αhj>0for all 0≤l≤s′ and all s′+1≤j≤M, for some integer 0≤s′≤M-1, hold.Remark 11.
In the caseq=1, k0,k1≥1, the roots of the polynomial (in t) P(t,ϵ)=a0ϵm0tk0+a1ϵm1tk1 all have modulus equal to (68)a1a01/k0-k1ϵm1-m0/k0-k1except the trivial root 0. The constraints (66) imply in particular that m1-m0>α(k1-k0). As a result, all the nonvanishing roots of P(t,ϵ) tend to ∞ as ϵ tends to 0 and 0 is therefore the only root (with order k0) of P(t,ϵ) in the vicinity of 0 as ϵ stays near the origin.
Let us assume that the expressionU(T,z,ϵ) is allowed to be written as a perturbation series with respect to ϵ:(69)UT,z,ϵ=U0T+∑n≥1UnT,zϵn,where the constant term U0(T) is taken independently of z and is not identically equal to 0. The coefficient U0(T) is called theslow curve of (64) in the terminology of [6].
In the following, we make the assumption thatU0(T) solves the following second-order algebraic equation:(70)∑l=0salTklU0T+∑l=0s′clThlU0T2=0.As U0(T) is not identically vanishing, it must be equal to -(∑l=0salTkl)/(∑l=0s′clThl). Bearing in mind that a0,c0≠0, we get its asymptotic behaviour(71)U0T~-a0c0Tk0-h0as T tends to 0.Remark 12.
Under the hypotheses (60) and (62), we observe, by factoring out the operator ∂zv from (64), that U(T,z,ϵ) must solve the related PDE(72)Q~∂z∑l=0qalϵml+β-αklTklUT,z,ϵ+∑l=0Mclϵμl+2β-αhlThlU2T,z,ϵ=∑j=0pb~jzϵnj-αbjTbj+FT,z,ϵ+∑l=1DϵΔl+αδl-dl+βTdlR~l∂z∂TδlUT,z,ϵ,where the forcing term F(T,z,ϵ) is a polynomial in z of degree less than v-1. According to the assumptions (65), (66), and (67) and using the fact that Q~(0)≠0, by taking ϵ=0 into (72), we see that the constraint (70) is equivalent to the fact that F(T,z,0)≡0. The precise shape of the term F(T,z,ϵ) will be given later in Section 6; see (183).
In afirst step, we express U(T,z,ϵ) as a small perturbation of U0(T) that can be expressed in the form (73)U0T=-a0c0Tk0-h0-a0c0Tk0-h0JT,where J(T)=∑j≥1JjTj is a convergent series near T=0; namely,(74)UT,z,ϵ=-a0c0Tk0-h0-a0c0Tk0-h0JT+TγVT,z,ϵfor some integer γ∈Z and some expression V(T,x,ϵ). By plugging this last expansion inside (64) and using the Leibniz rule, we get(75)Q∂z∑l=0salTkl+∑l=s+1qalϵml+β-αklTkl-a0c0Tk0-h0-a0c0Tk0-h0JT+TγVT,z,ϵ+∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl-a0c0Tk0-h0-a0c0Tk0-h0JT+TγVT,z,ϵ2=∑j=0pbjzϵnj-αbjTbj+∑l=1DϵΔl+αδl-dl+βTdlRl∂z-a0c0∏d=0δl-1k0-h0-dTk0-h0-δl-a0c0∑q1+q2=δlδl!q1!q2!∂Tq1Tk0-h0∂Tq2JT+∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dTγ-q1∂Tq2VT,z,ϵ,where we put ∏d=0-1(γ-d)=1 by convention. At this level, we observe the important fact that the coefficient in front of Q(∂z)V(T,z,ϵ) contains the term a0Tk0+γ-2(a0/c0)Tk0-h0+γc0Th0=-a0Tk0+γ that we want to set apart. As a result, we get the following equation satisfied by V(T,z,ϵ):(76)Q∂zVT,z,ϵ-a0Tk0+γ+∑l=1salTkl+∑l=s+1qalϵml+β-αklTklTγ-2a0c0Tk0-h0+γ∑l=1s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl-2a0c0Tk0-h0+γJT×∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl+Q∂zV2T,z,ϵT2γ∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl=∑j=0pbjzϵnj-αbjTbj+∑l=1DϵΔl+αδl-dl+β∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dTdl+γ-q1Rl∂z∂Tq2VT,z,ϵ.We now introduce some additional constraints on the integers γ, kl, hj, bh, for 0≤l≤q, 0≤j≤M, 0≤h≤p and dl, δl, for 1≤l≤D. Namely, we impose that the following inequalities hold:(77)k0-hp≤γfor 0≤p≤M, together with(78)γ≤bj-k0for all 0≤j≤p and finally(79)k0<dD-δD,k0≤dl-δlfor all 1≤l≤D-1.Remark 13.
(1) For the casek0>h0, from (77), we need γ≥k0-h0>0. As a consequence of (78), we get that bj>k0, for 0≤j≤p. Let, for instance, q=1, M=1, p=0, and D=2. We set α=2, β=1, γ=6, κ=1 and we choose the powers of t and ϵ in the coefficients of (61) as follows:(80)m0=3,k0=2,m1=6,k1=3,μ0=0,h0=1,μ1=3,h1=2,n0=19,b0=9,Δ1=12,d1=5,δ1=1,Δ2=20,d2=6,δ2=2.For these data, we can check that the constraints (65), (66), (67), (77), (78), and (79) above are fulfilled. Moreover, all the forthcoming requirements (88), (105), (106), (107), and (184) stated in Theorem 19 are also verified. In this special case, the main equation (61) writes(81)Q∂za0ϵ3t2+a1ϵ6t3ut,z,ϵ+c0t+c1ϵ3t2u2t,z,ϵ=b0zϵ19t9+ϵ12t5∂tR1∂zut,z,ϵ+ϵ20t6∂t2R2∂zut,z,ϵ.We can divide this last equation by t, but not by ϵ, and the resulting equation still possesses a turning point and an irregular singularity at t=0.
(2) For the caseh0>k0, we may take γ<0 and hence one can choose some bj<k0 for some 0≤j≤p. Let, for instance, q=1, M=1, p=0, and D=2. We choose α=2, β=-1, γ=-2, and κ=1 and we select the powers of t and ϵ in the coefficients of (61) as follows:(82)m0=5,k0=2,m1=10,k1=4,μ0=14,h0=6,μ1=19,h1=8,n0=3,b0=1,Δ1=10,d1=5,δ1=1,Δ2=12,d2=6,δ2=2.For these data, we can figure out that the constraints (65), (66), (67), (77), (78), and (79) above are satisfied. Moreover, all the forthcoming requirements (88), (105), (106), (107), and (184) stated in Theorem 19 are also verified. In this particular case, the main equation (61) writes(83)Q∂za0ϵ5t2+a1ϵ10t4ut,z,ϵ+c0ϵ14t6+c1ϵ19t8u2t,z,ϵ=b0zϵ3t+ϵ10t5∂tR1∂zut,z,ϵ+ϵ12t6∂t2R2∂zut,z,ϵ.We can divide this latter equation by ϵ3 and by t. The corresponding equation still suffers the presence of a turning point and an irregular singularity at t=0.
In asecond step, we divide the left- and right-hand sides of (76) by the monomial Tk0+γ. We obtain the following equation:(84)Q∂zVT,z,ϵ-a0+∑l=1salTkl+∑l=s+1qalϵml+β-αklTklT-k0-2a0c0T-h0∑l=1s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl-2a0c0T-h0JT×∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl+Q∂zV2T,z,ϵT-k0+γ∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl=∑j=0pbjzϵnj-αbjTbj-k0-γ+∑l=1DϵΔl+αδl-dl+β∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dTdl-k0-q1Rl∂z∂Tq2VT,z,ϵ.Notice that the additional constraints (65), (66), (67) and (77), (78), (79) ensure that the coefficients of the PDE (84) are analytic with respect to T and ϵ on a neighborhood of the origin in C2. Moreover, the coefficient of Q(∂z)V(T,z,ϵ) is invertible at T=0 since a0≠0. We will see later that this fact is essential in order to solve this equation within some function space of analytic functions.
We look for solutions which are rescaled in time of the form(85)VT,z,ϵ=VϵχT,z,ϵ,where(86)χ=ΔD+αδD-dD+βdD-k0-δD.As a result, provided that T=ϵχT holds, the expression V(T,z,ϵ) is supposed to solve the following equation:(87)Q∂zVT,z,ϵ-a0+∑l=1salϵ-χkl-k0Tkl-k0+∑l=s+1qalϵml+β-αkl-χkl-k0Tkl-k0-2a0c0∑l=1s′clϵ-χhl-h0Thl-h0+∑l=s′+1Mclϵμl+2β-αhl-χhl-h0Thl-h0-2a0c0Jϵ-χT×∑l=0s′clϵ-χhl-h0Thl-h0+∑l=s′+1Mclϵμl+2β-αhl-χhl-h0Thl-h0+Q∂zV2T,z,ϵϵ-χ-k0+γT-k0+γ∑l=0s′clϵ-χhlThl+∑l=s′+1Mclϵμl+2β-αhl-χhlThl=∑j=0pbjzϵnj-αbj-χbj-k0-γTbj-k0-γ+∑l=1D-1ϵΔl+αδl-dl+β×∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dϵ-χdl-k0-q1Tdl-k0-q1Rl∂zϵχq2∂Tq2VT,z,ϵ+ϵΔD+αδD-dD+β∑q1+q2=δD,q1≥1δD!q1!q2!∏d=0q1-1γ-dϵ-χdD-k0-q1TdD-k0-q1×RD∂zϵχq2∂Tq2VT,z,ϵ+TdD-k0RD∂z∂TδDVT,z,ϵ.
We make further assumptions on the coefficientsdl and δl for 1≤l≤D which are stronger than the constraint (79). Assume the existence of integers κ≥1 and dl,0≥1 such that(88)dD-k0=δDκ+1,dl-k0=δlκ+1+dl,0for all 1≤l≤D-1. Then, for all 1≤l≤D, and all integers q1≥0, q2≥0 with q1+q2=δl we deduce the existence of a nonnegative integer dl,q1,q2 which is larger than 1 except dD,0,δD=0 such that(89)dl-k0-q1=κ+1q2+dl,q1,q2.Indeed, if one puts dD,0=0, from (88), we can write(90)dl,q1,q2=dl-k0-q1-κ+1q2=δlκ+1+dl,0-q1-κ+1q2=q1+q2κ+1+dl,0-q1-κ+1q2=q1κ+dl,0.According to (88) and (89), with the help of formula (8.7) from [26], p. 3630, we can expand the following pieces appearing in (87) satisfied by V(T,z,ϵ):(91)TdD-k0∂TδDVT,z,ϵ=Tκ+1∂TδD+∑1≤p≤δD-1AδD,pTκδD-pTκ+1∂TpVT,z,ϵ,Tdl-k0-δl-1∂TVT,z,ϵ=Tdl,δl-1,1Tκ+1∂TVT,z,ϵ,Tdl-k0-q1∂Tq2VT,z,ϵ=Tdl,q1,q2Tκ+1q2∂Tq2VT,z,ϵ=Tdl,q1,q2Tκ+1∂Tq2+∑1≤p≤q2-1Aq2,pTκq2-pTκ+1∂TpVT,z,ϵfor all 1≤l≤D and all integers q1≥0 and q2≥2 such that q1+q2=δl, for some real constants AδD,p, 1≤p≤δD-1, and Aq2,p, 1≤p≤q2-1.
In athird step, let us assume that the expression V(T,z,ϵ) has a formal power series expansion(92)VT,z,ϵ=∑n≥1Vnz,ϵTn,where each coefficient Vn(z,ϵ) is defined as an inverse Fourier transform (93)Vnz,ϵ=F-1m⟼ωnm,ϵzfor some function m↦ωn(m,ϵ) belonging to the Banach space E(β,μ) and depending holomorphically on ϵ on some punctured disc D(0,ϵ0)∖{0} centered at 0 with radius ϵ0>0. We consider the formal power series (94)ωκτ,m,ϵ=∑n≥1ωnm,ϵΓn/κτnobtained by formally applying mκ-Borel transform with respect to T and Fourier transform with respect to z to the power series (92). The constraints (88) are introduced in such a way that ωκ(τ,m,ϵ) satisfies some integral equation by making use of the properties of the mκ-Borel transform of formal series and Fourier inverse transforms described in Propositions 8 and 9 with the help of the prepared expansions (91). Namely, after division by the power (im)v, which is by construction a common factor of the functions Q(im), Rl(im), and Bj(m) for 1≤l≤D, 0≤j≤p, we get the new problem(95)Lτ,m,ϵωκτ,m,ϵ=Rτ,m,ϵωκτ,m,ϵwith vanishing initial data ωκ(0,m,ϵ)≡0, where(96)Lτ,m,ϵωκτ,m,ϵ=Q~im-a0ωκτ,m,ϵ+∑l=1salϵ-χkl-k0τκΓkl-k0/κ∫0τκτκ-skl-k0/κ-1ωκs1/κ,m,ϵdss+∑l=s+1qalϵml+β-αkl-χkl-k0τκΓkl-k0/κ∫0τκτκ-skl-k0/κ-1ωκs1/κ,m,ϵdss-2a0c0∑l=1s′clϵ-χhl-h0τκΓhl-h0/κ∫0τκτκ-shl-h0/κ-1ωκs1/κ,m,ϵdss+∑l=s′+1Mclϵμl+2β-αhlϵ-χhl-h0τκΓhl-h0/κ∫0τκτκ-shl-h0/κ-1ωκs1/κ,m,ϵdss-2a0c0∑l=0s′cl∑j≥1Jjϵ-χhl-h0+jτκΓhl-h0+j/κ∫0τκτκ-shl-h0+j/κ-1ωκs1/κ,m,ϵdss+∑l=s′+1Mcl∑j≥1Jjϵμl+2β-αhlϵ-χhl-h0+jτκΓhl-h0+j/κ∫0τκτκ-shl-h0+j/κ-1ωκs1/κ,m,ϵdss+Q~im∑l=0s′clϵ-χ-k0+γ+hlτκΓ-k0+γ+hl/κ∫0τκτκ-s-k0+γ+hl/κ-1×s∫0s∫-∞+∞12π1/2ωκs-s′1/κ,m-m1,ϵωκs′1/κ,m1,ϵ1s-s′s′ds′dm1dss+∑l=s′+1Mclϵμl+2β-αhl-χhl-k0+γτκΓ-k0+γ+hl/κ∫0τκτκ-s-k0+γ+hl/κ-1×s∫0s∫-∞+∞12π1/2ωκs-s′1/κ,m-m1,ϵωκs′1/κ,m1,ϵ1s-s′s′ds′dm1dss,(97)Rτ,m,ϵωκτ,m,ϵ=∑j=0pB~jmϵnj-αbj-χbj-k0-γτbj-k0-γΓbj-k0-γ/κ+∑l=1D-1ϵΔl+αδl-dl+β×∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dϵ-χdl-k0-q1-q2R~lim×τκΓdl,q1,q2/κ∫0τκτκ-sdl,q1,q2/κ-1κq2sq2ωκs1/κ,m,ϵdss+∑1≤p≤q2-1Aq2,pτκΓdl,q1,q2+κq2-p/κ∫0τκτκ-sdl,q1,q2+κq2-p/κ-1κpspωκs1/κ,m,ϵdss+ϵΔD+αδD-dD+β×∑q1+q2=δD,q1≥1δD!q1!q2!∏d=0q1-1γ-dϵ-χdD-k0-q1-q2R~Dim×τκΓdD,q1,q2/κ∫0τκτκ-sdD,q1,q2/κ-1κq2sq2ωκs1/κ,m,ϵdss+∑1≤p≤q2-1Aq2,pτκΓdD,q1,q2+κq2-p/κ∫0τκτκ-sdD,q1,q2+κq2-p/κ-1κpspωκs1/κ,m,ϵdss+R~DimκτκδDωκτ,m,ϵ+∑1≤p≤δD-1AδD,pτκΓκδD-p/κ×∫0τκτκ-sκδD-p/κ-1κpspωκs1/κ,m,ϵdss.By convention, the two sums ∑1≤p≤q2-1[⋯] appearing in (97) are vanishing provided that q2∈{0,1}.Remark 14.
The hypotheses (60) and (62) ensure that (87) does not contain terms that involve isolated polynomials in T which are not inverse Fourier transformable.


## 5. Analytic Solutions of a Convolution Problem with Complex Parameters

Our main goal in this section is the construction of a unique solution of problem (95) within the Banach spaces introduced in Section 2.We make the following further assumptions. The conditions below are very similar to the ones proposed in Section  4 of [3]. Namely, we demand that there exists an unbounded sector (98)SQ~,R~D=z∈C∣z≥rQ~,R~D,argz-dQ~,R~D≤ηQ~,R~Dwith direction dQ~,R~D∈R and aperture ηQ~,R~D>0 for some radius rQ~,R~D>0 such that(99)Q~imR~Dim∈SQ~,R~Dfor all m∈R. The polynomial P~m(τ)=-Q~(im)a0-R~D(im)κδDτδDκ can be factorized in the form(100)P~mτ=-R~DimκδD∏l=0δDκ-1τ-qlm,where(101)qlm=a0Q~imR~DimκδD1/δDκexp⁡-1arg⁡-a0Q~imR~DimκδD1δDκ+2πlδDκfor all 0≤l≤δDκ-1 and all m∈R.We select an unbounded sectorSd centered at 0 and a small closed disc D¯(0,ρ) and we require the sector SQ~,R~D to fulfill the next conditions.(1) There exists a constantM1>0 such that(102)τ-qlm≥M11+τfor all 0≤l≤δDκ-1, all m∈R, and all τ∈Sd∪D¯(0,ρ). Indeed, from (99) and the explicit expression (101) of ql(m), we first observe that qlm>2ρ for every m∈R, all 0≤l≤δDκ-1 for an appropriate choice of rQ~,R~D and of ρ>0. We also see that, for all m∈R and all 0≤l≤δDκ-1, the roots ql(m) remain in a union U of unbounded sectors centered at 0 that do not cover a full neighborhood of the origin in C∗ provided that ηQ~,R~D is small enough. Therefore, one can choose an adequate sector Sd such that Sd∩U=∅ with the property that for all 0≤l≤δDκ-1 the quotients ql(m)/τ lay outside some small disc centered at 1 in C for all τ∈Sd and all m∈R. This yields (102) for some small constant M1>0.(2) There exists a constantM2>0 such that(103)τ-ql0m≥M2ql0mfor some l0∈{0,…,δDκ-1}, all m∈R, and all τ∈Sd∪D¯(0,ρ). Indeed, for the sector Sd and the disc D¯(0,ρ) chosen as above in (1), we notice that for any fixed 0≤l0≤δDκ-1, the quotient τ/ql0(m) stays outside a small disc centered at 1 in C for all τ∈Sd∪D¯(0,ρ) and all m∈R. Hence (103) must hold for some small constant M2>0.By construction of the roots (101) in the factorization (100) and using the lower bound estimates (102) and (103), we get a constant CP~>0 such that(104)P~mτ≥M1δDκ-1M2R~DimκδDa0Q~imR~DimκδD1/δDκ1+τδDκ-1≥M1δDκ-1M2κδDa01/δDκκδD1/δDκrQ~,R~D1/δDκR~Dim×minx≥0⁡1+xδDκ-11+xκδD-1/κ1+τκδD-1/κ=CP~rQ~,R~D1/δDκR~Dim1+τκδD-1/κfor all τ∈Sd∪D¯(0,ρ) and all m∈R.In the next proposition, we provide sufficient conditions under which the main convolution equation (95) possesses solutions ωκd(τ,m,ϵ) in the Banach space F(ν,β,μ,χ,α,κ,ϵ)d described in Section 2.Proposition 15.
Under the additional assumptions(105)χ+α-k0+γ+hl-κδD+1-χ-k0+γ+hl≥0,δD≥2κ,-k0+γ+hl>0,bj-k0-γ≥1,for all 0≤l≤M, 0≤j≤p,(106)Δl+αδl-dl+β+χ+ακdl,q1,q2κ+q2-δD+1κ-χdl-k0-δl≥0,δD≥1κ+δlfor all q1≥0, q2≥1 such that q1+q2=δl, for 1≤l≤D-1 and(107)ΔD+αδD-dD+β+χ+ακdD,q1,q2κ+q2-δD+1κ-χdD-k0-δD≥0for all q1≥1, q2≥1 such that q1+q2=δD, there exist a radius rQ~,R~D>0, ϵ0>0 and a constant ϖ>0 such that (95) has a unique solution ωκd(τ,m,ϵ) in the Banach space F(ν,β,μ,χ,α,κ,ϵ)d which suffers the bounds (108)ωκdτ,m,ϵν,β,μ,χ,α,κ,ϵ≤ϖfor all ϵ∈D0,ϵ0∖{0}, where the direction d∈R can be chosen for any sector Sd that fulfills the constraints (102) and (103) above.Proof. We undertake the proof with a lemma that studies some shrinking map on the Banach spaces mentioned above and reduces the main convolution problem (95) to the existence of a unique fixed point for this map.Lemma 16.
Taking for granted the fact that the assumptions (105), (106), and (107) hold, one can select the constant rQ~,R~D>0 large enough and a constant ϖ>0 small enough such that for all ϵ∈D0,ϵ0∖{0}, the map Hϵ defined as(109)Hϵ=Hϵ1+Hϵ2+Hϵ3,where(110)Hϵ1wτ,m≔∑j=0pB~jmϵnj-αbj-χbj-k0-γτbj-k0-γP~mτΓbj-k0-γ/κ-Q~im∑l=1salϵ-χkl-k0τκP~mτΓkl-k0/κ∫0τκτκ-skl-k0/κ-1ws1/κ,mdss+∑l=s+1qalϵml+β-αkl-χkl-k0τκP~mτΓkl-k0/κ∫0τκτκ-skl-k0/κ-1ws1/κ,mdss-2a0c0∑l=1s′clϵ-χhl-h0τκP~mτΓhl-h0/κ∫0τκτκ-shl-h0/κ-1ws1/κ,mdss+∑l=s′+1Mclϵμl+2β-αhlϵ-χhl-h0τκP~mτΓhl-h0/κ∫0τκτκ-shl-h0/κ-1ws1/κ,mdss-2a0c0∑l=0s′cl∑j≥1Jjϵ-χhl-h0+jτκP~mτΓhl-h0+j/κ∫0τκτκ-shl-h0+j/κ-1ws1/κ,mdss+∑l=s′+1Mcl∑j≥1Jjϵμl+2β-αhlϵ-χhl-h0+j×τκP~mτΓhl-h0+j/κ∫0τκτκ-shl-h0+j/κ-1ws1/κ,mdss,(111)Hϵ2wτ,m≔-Q~im∑l=0s′clϵ-χ-k0+γ+hlτκP~mτΓ-k0+γ+hl/κ∫0τκτκ-s-k0+γ+hl/κ-1×s∫0s∫-∞+∞12π1/2ws-s′1/κ,m-m1ws′1/κ,m11s-s′s′ds′dm1dss+∑l=s′+1Mclϵμl+2β-αhl-χhl-k0+γτκP~mτΓ-k0+γ+hl/κ∫0τκτκ-s-k0+γ+hl/κ-1×s∫0s∫-∞+∞12π1/2ws-s′1/κ,m-m1ws′1/κ,m11s-s′s′ds′dm1dss+∑l=1D-1ϵΔl+αδl-dl+β×∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dϵ-χdl-k0-q1-q2R~lim×τκP~mτΓdl,q1,q2/κ∫0τκτκ-sdl,q1,q2/κ-1κq2sq2ws1/κ,mdss+∑1≤p≤q2-1Aq2,pτκP~mτΓdl,q1,q2+κq2-p/κ∫0τκτκ-sdl,q1,q2+κq2-p/κ-1κpspws1/κ,mdss,along with(112)Hϵ3wτ,m≔ϵΔD+αδD-dD+β×∑q1+q2=δD,q1≥1δD!q1!q2!∏d=0q1-1γ-dϵ-χdD-k0-q1-q2R~Dim×τκP~mτΓdD,q1,q2/κ∫0τκτκ-sdD,q1,q2/κ-1κq2sq2ws1/κ,mdss+∑1≤p≤q2-1Aq2,pτκP~mτΓdD,q1,q2+κq2-p/κ∫0τκτκ-sdD,q1,q2+κq2-p/κ-1κpspws1/κ,mdss+R~Dim∑1≤p≤δD-1AδD,pτκP~mτΓκδD-p/κ×∫0τκτκ-sκδD-p/κ-1κpspws1/κ,mdss,satisfies the next properties.
(i) The following inclusion holds:(113)HϵB¯0,ϖ⊂B¯0,ϖ,where B¯(0,ϖ) is the closed ball of radius ϖ>0 centered at 0 in F(ν,β,μ,χ,α,κ,ϵ)d, for all ϵ∈D(0,ϵ0)∖{0}.
(ii) One has(114)Hϵw1-Hϵw2ν,β,μ,χ,α,κ,ϵ≤12w1-w2ν,β,μ,χ,α,κ,ϵfor all w1,w2∈B¯(0,ϖ), for all ϵ∈D(0,ϵ0)∖{0}.Proof.
We first deal with the property (113). Let ϵ∈D(0,ϵ0)∖{0} and consider w(τ,m)∈F(ν,β,μ,χ,α,κ,ϵ)d. We take ϖ>0 such that w(τ,m)(ν,β,μ,χ,α,κ,ϵ)≤ϖ.
We start providing norm estimates for each piece of the mapHϵ1.
From Lemma4, we deduce the existence of a constant C1>0 depending on κ, γ, k0, and bj for 0≤j≤p such that(115)B~jmϵ-χbj-k0-γτbj-k0-γP~mτν,β,μ,χ,α,κ,ϵ≤C1CP~rQ~,R~D1/δDκB~jmβ,μinfm∈RR~Dimϵbj-k0-γα.According to Proposition 5(1), we obtain a constant C2>0 depending on ν,κ, kl, for 0≤l≤q, hl, for 0≤l≤M, Q~(X), R~D(X), and a constant C2(j) depending on ν, κ, hl for 0≤l≤M, Q~(X), R~DX and j such that(116)ϵ-χkl-k0Q~imτκP~mτ∫0τκτκ-skl-k0/κ-1ws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2CP~rQ~,R~D1/δDκϵαkl-k0wτ,mν,β,μ,χ,α,κ,ϵ,ϵ-χhl-h0Q~imτκP~mτ∫0τκτκ-shl-h0/κ-1ws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2CP~rQ~,R~D1/δDκϵαhl-h0wτ,mν,β,μ,χ,α,κ,ϵ,ϵ-χhl-h0+jQ~imP~mττκ∫0τκτκ-shl-h0+j/κ-1ws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2jCP~rQ~,R~D1/δDκϵαhl-h0+jwτ,mν,β,μ,χ,α,κ,ϵ.Moreover, it appears from the proof of Proposition 5 that the next bounds hold for C2(j): there exist a constant C^2>0 depending on ν, κ, hl for 0≤l≤M, Q~, R~D and a constant A2>0 depending on ν, κ, hl for 0≤l≤M such that (117)C2j≤C^2A2jΓhl-h0+jκfor all j≥1. In the following, we will make use of the notations from the proof of Proposition 5. From the classical estimates(118)supx≥0⁡xm1e-m2x=m1m2m1e-m1for any real numbers m1≥0, m2>0, we deduce that for all j≥1 such that (hl-h0+j)/κ-1>0(119)supx≥0⁡1+x2x1/κe-νxxG1x≤supx≥0⁡1+x2xhl-h0+j/κe-ν/2x121/κκ≤hl-h0+jκν/2hl-h0+j/κexp⁡-hl-h0+jκ+hl-h0+j/κ+2ν/2hl-h0+j/κ+2exp⁡-hl-h0+jκ+2121/κκ.Furthermore, according to the Stirling formula Γx~2πxx-1/2e-x as x→+∞ and bearing in mind the functional relation Γx+1=xΓ(x) for all x>0, we get two constants Cˇ2>0 and A2>0 independent of j such that(120)supx≥0⁡1+x2x1/κe-νxxG1x≤Cˇ2A2jΓhl-h0+jκ+Γhl-h0+jκ+2≤Cˇ2A2jΓhl-h0+jκ+hl-h0+jκ+1hl-h0+jκΓhl-h0+jκ.On the other hand, by direct inspection, we observe that there exists a constant Cˇ2.1>0 (independent of j and ϵ) such that(121)sup0≤x<1⁡1+x2x1/κe-νxx1+ϵχ+ακxγ1G2x≤Cˇ2.1.Furthermore, there exists a constant K2.1(j) depending on ν, κ, hl for 0≤l≤M and j, such that(122)supx≥1⁡1+x2x1/κe-νxx1+ϵχ+ακxγ1G2x≤supx≥1⁡1+x21+x/22K2.1j.Now, after a thorough examination of the proof of Proposition 2 out of [23], one can check that there exists a constant Kˇ2.1>0 independent of j such that(123)K2.1j≤Kˇ2.1Γhl-h0+jκfor all j≥1. Finally, gathering (120), (121), (122), and (123) yields the estimates (117).
Besides, we choose the radiusrQ~,R~D>0 large enough and ϖ in such a manner that(124)∑j=0pC1CP~rQ~,R~D1/δDκΓbj-k0-γ/κB~jmβ,μinfm∈R⁡R~Dimϵnj-αbj+bj-k0-γα+∑l=1salC2CP~rQ~,R~D1/δDκΓkl-k0/κϵαkl-k0ϖ+∑l=s+1qalC2CP~rQ~,R~D1/δDκΓkl-k0/κϵml+β-αkl+αkl-k0ϖ+2a0c0∑l=1s′clC2CP~rQ~,R~D1/δDκΓhl-h0/κϵαhl-h0ϖ+∑l=s′+1MclC2CP~rQ~,R~D1/δDκΓhl-h0/κϵμl+2β-αhlϵαhl-h0ϖ+2a0c0∑l=0s′cl∑j≥1JjC^2A2jCP~rQ~,R~D1/δDκϵαhl-h0+jϖ+∑l=s′+1Mcl∑j≥1JjC^2A2jCP~rQ~,R~D1/δDκϵμl+2β-αhlϵαhl-h0+jϖ≤ϖ3.Notice that the infinite sums over the integers j≥0 are convergent in the left-hand side of inequality (124), provided that ϵ0>0 is small enough, according to the fact that there exist two constants J1,J2>0 such that Jj≤J1(J2)j for all j≥1 since J(T)=∑j≥1JjTj is a convergent series near T=0.
From the definition ofHϵ1 given by (110), we deduce the following inequality:(125)Hϵ1wτ,mν,β,μ,χ,α,κ,ϵ≤ϖ3.Hereafter, we focus on norm estimates for each part of the map Hϵ2.
We set(126)hτ,m=τκ-1∫0τκ∫-∞+∞wτκ-s′1/κ,m-m1ws′1/κ,m11τκ-s′s′ds′dm1.Regarding Proposition 6, we get a constant C3>0 (depending on μ, κ) such that(127)hτ,mν,β,μ,χ,α,κ,ϵ≤C3ϵχ+αwτ,mν,β,μ,χ,α,κ,ϵ2.On the other hand, using Proposition 5(2), we grab a constant C2′>0 (depending on ν, κ, γ, δD, k0, hl for 0≤l≤M and Q~(X), R~D(X)) such that(128)ϵ-χ-k0+γ+hlQ~imP~mττκ∫0τκτκ-s-k0+γ+hl/κ-1s1/κ-1hs1/κ,mdsν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακ-k0+γ+hl/κ+1/κ-χ-k0+γ+hl-χ+ακδD-1/κhτ,mν,β,μ,χ,α,κ,ϵ.Therefore, gathering (127) and (128) returns(129)ϵ-χ-k0+γ+hlQ~imP~mττκ∫0τκτκ-s-k0+γ+hl/κ-1×s∫0s∫-∞+∞ws-s′1/κ,m-m1ws′1/κ,m11s-s′s′ds′dm1dssν,β,μ,χ,α,κ,ϵ≤C2′C3CP~rQ~,R~D1/δDκϵχ+α-k0+γ+hl-κδD+1-χ-k0+γ+hlwτ,mν,β,μ,χ,α,κ,ϵ2.Bearing in mind Proposition 5(1), we get a constant C2>0 (depending on ν, κ, dl, δl and R~l(X), R~D(X) for 1≤l≤D-1), such that(130)ϵ-χdl-k0-δlR~limP~mττκ∫0τκτκ-sdl,δl,0/κ-1ws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2CP~rQ~,R~D1/δDκϵχ+αdl,δl,0-χdl-k0-δlwτ,mν,β,μ,χ,α,κ,ϵ=C2CP~rQ~,R~D1/δDκϵαdl-k0-δlwτ,mν,β,μ,χ,α,κ,ϵ.Likewise, we can apply Proposition 5(2) in order to exhibit a constant C2′>0 (depending on ν, κ, dl, δl, k0, δD and R~l(X), R~D(X) for 1≤l≤D-1) with(131)ϵ-χdl-k0-δlR~limP~mττκ∫0τκτκ-sdl,q1,q2/κ-1sq2ws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακdl,q1,q2/κ+q2-δD+1/κ-χdl-k0-δlwτ,mν,β,μ,χ,α,κ,ϵfor all q1≥0 and q2≥1 with q1+q2=δl. Besides,(132)ϵ-χdl-k0-δlR~limP~mττκ∫0τκτκ-sdl,q1,q2/κ+q2-p-1spws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακdl,q1,q2/κ+q2-δD+1/κ-χdl-k0-δlwτ,mν,β,μ,χ,α,κ,ϵprovided that q2≥2 and 1≤p≤q2-1, with q1+q2=δl.
Now, we chooserQ~,R~D>0 and ϖ in such a way that(133)∑l=0s′clC2′C3CP~rQ~,R~D1/δDκΓ-k0+γ+hl/κ2π1/2ϵχ+α-k0+γ+hl-κδD+1-χ-k0+γ+hlϖ2+∑l=s′+1MclC2′C3CP~rQ~,R~D1/δDκΓ-k0+γ+hl/κ2π1/2×ϵχ+α-k0+γ+hl-κδD+1-χ-k0+γ+hlϵμl+2β-αhlϖ2+∑l=1D-1ϵΔl+αδl-dl+β∏d=0δl-1γ-dC2CP~rQ~,R~D1/δDκΓdl,δl,0/κϵαdl-k0-δlϖ+∑q1+q2=δl,q2≥1δl!q1!q2!∏d=0q1-1γ-dC2′κq2CP~rQ~,R~D1/δDκΓdl,q1,q2/κ×ϵχ+ακdl,q1,q2/κ+q2-δD+1/κ-χdl-k0-δlϖ+∑1≤p≤q2-1Aq2,pC2′κpCP~rQ~,R~D1/δDκΓdl,q1,q2/κ+q2-p×ϵχ+ακdl,q1,q2/κ+q2-δD+1/κ-χdl-k0-δlϖ≤ϖ3.With the help of the definition of Hϵ2 given by (111), we deduce that (134)Hϵ2wτ,mν,β,μ,χ,α,κ,ϵ≤ϖ3.Ultimately, we direct our attention to norm estimates for Hϵ3.
Taking notice of Proposition5(1), we get a constant C2>0 (depending on ν, κ, k0, δD, dD), such that(135)ϵ-χdD-k0-δDR~DimP~mττκ∫0τκτκ-sdD,δD,0/κ-1ws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2CP~rQ~,R~D1/δDκϵχ+αdD,δD,0-χdD-k0-δDwτ,mν,β,μ,χ,α,κ,ϵ=C2CP~rQ~,R~D1/δDκϵαdD-k0-δDwτ,mν,β,μ,χ,α,κ,ϵ.Moreover, we can apply Proposition 5(2) in order to exhibit a constant C2′>0 (depending on ν, κ, k0, dD, and δD) with(136)ϵ-χdD-k0-δDR~DimP~mττκ∫0τκτκ-sdD,q1,q2/κ-1sq2ws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακdD,q1,q2/κ+q2-δD+1/κ-χdD-k0-δDwτ,mν,β,μ,χ,α,κ,ϵfor all q1≥1 and q2≥1 with q1+q2=δD. Besides,(137)ϵ-χdD-k0-δDR~DimP~mττκ∫0τκτκ-sdD,q1,q2/κ+q2-p-1spws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακdD,q1,q2/κ+q2-δD+1/κ-χdD-k0-δDwτ,mν,β,μ,χ,α,κ,ϵprovided that q1≥1, q2≥2, and 1≤p≤q2-1 with q1+q2=δD. Finally, we can select a constant C2′>0 (depending on ν, κ, and δD) such that(138)R~DimP~mττκ∫0τκτκ-sδD-p-1spws1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+αwτ,mν,β,μ,χ,α,κ,ϵfor all 1≤p≤δD-1. We make the choice for the size of radius rQ~,R~D and ϖ in such a manner that(139)ϵΔD+αδD-dD+β∏d=0δD-1γ-dC2CP~rQ~,R~D1/δDκΓdD,δD,0/κϵαdD-k0-δDϖ+∑q1+q2=δD,q1≥1,q2≥1δD!q1!q2!∏d=0q1-1γ-dC2′κq2CP~rQ~,R~D1/δDκΓdD,q1,q2/κ×ϵχ+ακdD,q1,q2/κ+q2-δD+1/κ-χdD-k0-δDϖ+∑1≤p≤q2-1Aq2,pC2′κpCP~rQ~,R~D1/δDκΓdD,q1,q2/κ+q2-p×ϵχ+ακdD,q1,q2/κ+q2-δD+1/κ-χdD-k0-δDϖ+∑1≤p≤δD-1AδD,pC2′κpCP~rQ~,R~D1/δDκΓδD-pϵχ+αϖ≤ϖ3.From the construction of the map Hϵ3, it is now clear that(140)Hϵ3wτ,mν,β,μ,χ,α,κ,ϵ≤ϖ3.Eventually, gathering (125), (134), and (140) yields the first claim (113).
In the last part of the proof, we fix our attention to the affirmation (114). Let w1(τ,m),w2(τ,m)∈F(ν,β,μ,χ,α,κ,ϵ)d with (141)w1τ,mν,β,μ,χ,α,κ,ϵ≤ϖ,w2τ,mν,β,μ,χ,α,κ,ϵ≤ϖ.We first prove that Hϵ1 is a shrinking map. According to estimates (116) we obtain a constant C2>0 (depending on ν, κ, kl, for 0≤l≤q, hl, for 0≤l≤M, Q~(X), R~D(X)) and a constant C2(j) satisfying the estimates (117) such that(142)ϵ-χkl-k0Q~imτκP~mτ∫0τκτκ-skl-k0/κ-1w1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2CP~rQ~,R~D1/δDκϵαkl-k0w1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ,ϵ-χhl-h0Q~imτκP~mτ∫0τκτκ-shl-h0/κ-1w1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2CP~rQ~,R~D1/δDκϵαhl-h0w1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ,ϵ-χhl-h0+jQ~imP~mττκ∫0τκτκ-shl-h0+j/κ-1w1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2jCP~rQ~,R~D1/δDκϵαhl-h0+jw1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ.Therefore, we choose the radius rQ~,R~D>0 large enough in order that(143)∑l=1salC2CP~rQ~,R~D1/δDκΓkl-k0/κϵαkl-k0+∑l=s+1qalC2CP~rQ~,R~D1/δDκΓkl-k0/κϵml+β-αkl+αkl-k0+2a0c0∑l=1s′clC2CP~rQ~,R~D1/δDκΓhl-h0/κϵαhl-h0+∑l=s′+1MclC2CP~rQ~,R~D1/δDκΓhl-h0/κϵμl+2β-αhlϵαhl-h0+2a0c0∑l=0s′cl∑j≥1JjC^2A2jCP~rQ~,R~D1/δDκϵαhl-h0+j+∑l=s′+1Mcl∑j≥1JjC^2A2jCP~rQ~,R~D1/δDκϵμl+2β-αhlϵαhl-h0+j≤16.As a result, we can set down(144)Hϵ1w1τ,m-Hϵ1w2τ,mν,β,μ,χ,α,κ,ϵ≤16w1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ.We turn to Hϵ2 and show that it is a shrinking map as well. As a preparation, we may first rewrite(145)w1τκ-s′1/κ,m-m1w1s′1/κ,m1-w2τκ-s′1/κ,m-m1w2s′1/κ,m1=w1τκ-s′1/κ,m-m1-w2τκ-s′1/κ,m-m1w1s′1/κ,m1+w2τκ-s′1/κ,m-m1w1s′1/κ,m1-w2s′1/κ,m1.For j=1,2, we set (146)hjτ,m=τκ-1∫0τκ∫-∞+∞wjτκ-s′1/κ,m-m1wjs′1/κ,m11τκ-s′s′ds′dm1.Regarding both the factorization (145) above and Proposition 6, we get a constant C3>0 (depending on μ, κ) such that(147)h1τ,m-h2τ,mν,β,μ,χ,α,κ,ϵ≤C3ϵχ+αw1τ,mν,β,μ,χ,α,κ,ϵ+w2τ,mν,β,μ,χ,α,κ,ϵ×w1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ.From (128) together with (147) we pick up a constant C2′>0 (depending on ν, κ, γ, δD, k0, hl for 0≤l≤M and Q~(X), R~D(X)) such that(148)ϵ-χ-k0+γ+hlQ~imP~mττκ∫0τκτκ-s-k0+γ+hl/κ-1s1/κ-1h1s1/κ,m-h2s1/κ,mdsν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακ-k0+γ+hl/κ+1/κ-χ-k0+γ+hl-χ+ακδD-1/κh1τ,m-h2τ,mν,β,μ,χ,α,κ,ϵ≤C2′C3CP~rQ~,R~D1/δDκϵχ+α-k0+γ+hl-κδD+1-χ-k0+γ+hl×w1τ,mν,β,μ,χ,α,κ,ϵ+w2τ,mν,β,μ,χ,α,κ,ϵw1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ.Bearing in mind (130), we get a constant C2>0 (depending on ν, κ, dl, δl and R~l(X), R~D(X) for 1≤l≤D-1), such that(149)ϵ-χdl-k0-δlR~limP~mττκ∫0τκτκ-sdl,δl,0/κ-1w1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2CP~rQ~,R~D1/δDκϵαdl-k0-δlw1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ.Likewise, we can apply (131) in order to exhibit a constant C2′>0 (depending on ν, κ, dl, δl, k0, δD and R~l(X), R~D(X) for 1≤l≤D-1) with(150)ϵ-χdl-k0-δlR~limP~mττκ∫0τκτκ-sdl,q1,q2/κ-1sq2w1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακdl,q1,q2/κ+q2-δD+1/κ-χdl-k0-δlw1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵfor all q1≥0 and q2≥1 with q1+q2=δl. Furthermore, from (132) we deduce(151)ϵ-χdl-k0-δlR~limP~mττκ∫0τκτκ-sdl,q1,q2/κ+q2-p-1spw1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακdl,q1,q2/κ+q2-δD+1/κ-χdl-k0-δlw1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵprovided that q2≥2 and 1≤p≤q2-1, with q1+q2=δl.
Now, we sortrQ~,R~D>0 and ϖ in such a way that(152)∑l=0s′clC2′C3CP~rQ~,R~D1/δDκΓ-k0+γ+hl/κ2π1/2ϵχ+α-k0+γ+hl-κδD+1-χ-k0+γ+hl×w1τ,mν,β,μ,χ,α,κ,ϵ+w2τ,mν,β,μ,χ,α,κ,ϵ+∑l=s′+1MclC2′C3CP~rQ~,R~D1/δDκΓ-k0+γ+hl/κ2π1/2ϵχ+α-k0+γ+hl-κδD+1-χ-k0+γ+hl×ϵμl+2β-αhlw1τ,mν,β,μ,χ,α,κ,ϵ+w2τ,mν,β,μ,χ,α,κ,ϵ+∑l=1D-1ϵΔl+αδl-dl+β∏d=0δl-1γ-dC2CP~rQ~,R~D1/δDκΓdl,δl,0/κϵαdl-k0-δl+∑q1+q2=δl,q2≥1δl!q1!q2!∏d=0q1-1γ-dC2′κq2CP~rQ~,R~D1/δDκΓdl,q1,q2/κ×ϵχ+ακdl,q1,q2/κ+q2-δD+1/κ-χdl-k0-δl+∑1≤p≤q2-1Aq2,pC2′κpCP~rQ~,R~D1/δDκΓdl,q1,q2/κ+q2-p×ϵχ+ακdl,q1,q2/κ+q2-δD+1/κ-χdl-k0-δl≤16.Subsequently, we obtain(153)Hϵ2w1τ,m-Hϵ2w2τ,mν,β,μ,χ,α,κ,ϵ≤16w1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ.The last operation will be devoted to the proof that Hϵ3 is a shrinking map.
Taking notice of (135), we get a constant C2>0 (depending on ν, κ, k0, δD, and dD), such that(154)ϵ-χdD-k0-δDR~DimP~mττκ∫0τκτκ-sdD,δD,0/κ-1w1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2CP~rQ~,R~D1/δDκϵαdD-k0-δDw1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ.Moreover, we may have a look at (136) in order to exhibit a constant C2′>0 (depending on ν, κ, k0, dD, and δD) with(155)ϵ-χdD-k0-δDR~DimP~mττκ×∫0τκτκ-sdD,q1,q2/κ-1sq2w1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακdD,q1,q2/κ+q2-δD+1/κ-χdD-k0-δDw1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵfor all q1≥1 and q2≥1 with q1+q2=δD. Besides, from (137), we see that(156)ϵ-χdD-k0-δDR~DimP~mττκ×∫0τκτκ-sdD,q1,q2/κ+q2-p-1spw1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+ακdD,q1,q2/κ+q2-δD+1/κ-χdD-k0-δDw1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵprovided that q1≥1, q2≥2, and 1≤p≤q2-1 with q1+q2=δD. Finally, having a glance at (138), we can select a constant C2′>0 (depending on ν, κ, and δD) such that(157)R~DimP~mττκ∫0τκτκ-sδD-p-1spw1s1/κ,m-w2s1/κ,mdssν,β,μ,χ,α,κ,ϵ≤C2′CP~rQ~,R~D1/δDκϵχ+αw1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵfor all 1≤p≤δD-1. In the meanwhile, we select the size of radius rQ~,R~D in such a manner that(158)ϵΔD+αδD-dD+β∏d=0δD-1γ-dC2CP~rQ~,R~D1/δDκΓdD,δD,0/κϵαdD-k0-δD+∑q1+q2=δD,q1≥1,q2≥1δD!q1!q2!∏d=0q1-1γ-dC2′κq2CP~rQ~,R~D1/δDκΓdD,q1,q2/κ×ϵχ+ακdD,q1,q2/κ+q2-δD+1/κ-χdD-k0-δD+∑1≤p≤q2-1Aq2,pC2′κpCP~rQ~,R~D1/δDκΓdD,q1,q2/κ+q2-p×ϵχ+ακdD,q1,q2/κ+q2-δD+1/κ-χdD-k0-δD+∑1≤p≤δD-1AδD,pC2′κpCP~rQ~,R~D1/δDκΓδD-pϵχ+α≤16.The following inequality must then hold:(159)Hϵ3w1τ,m-Hϵ3w2τ,mν,β,μ,χ,α,κ,ϵ≤16w1τ,m-w2τ,mν,β,μ,χ,α,κ,ϵ.Gathering (144), (153), and (159) legitimates estimates (114).
At the very end of the proof, we now take for granted that all conditions (124), (133), (139), (143), (152), and (158) hold for the radii rQ~,R~D and ϖ. Then both (113) and (114) hold at the same time and Lemma 16 is shown.We consider the closed ballB¯(0,ϖ) just built above in Lemma 16 which is actually a complete metric space with respect to the metric induced by the Banach space norm ·(ν,β,μ,χ,α,κ,ϵ). From the lemma above, we get that Hϵ is a contractive map from B¯(0,ϖ) into itself. Due to the classical contractive mapping theorem, we deduce that the map Hϵ has a unique fixed point denoted by ωκd(τ,m,ϵ) in the ball B¯(0,ϖ), meaning that(160)Hϵωκdτ,m,ϵ=ωκdτ,m,ϵfor a unique ωκd(τ,m,ϵ)∈F(ν,β,μ,χ,α,κ,ϵ)d such that ωκd(τ,m,ϵ)(ν,β,μ,χ,α,κ,ϵ)≤ϖ, for all ϵ∈D(0,ϵ0)∖{0}. Moreover, the function ωκd(τ,m,ϵ) depends holomorphically on ϵ in D(0,ϵ0)∖{0}.Now, if one sets apart the terms-a0Q~(im)ωκ(τ,m,ϵ) in the left-hand side Lτ,m,ϵ and R~D(im)(κτκ)δDωκ(τ,m,ϵ) in the right-hand side Rτ,m,ϵ of (95), we observe by dividing with the polynomial P~m(τ) given in (100) that (95) can be exactly rewritten as (160). Therefore, the unique fixed point ωκd(τ,m,ϵ) of Hϵ in B¯(0,ϖ) precisely solves problem (95) with vanishing initial data ωκ(0,m,ϵ)≡0. This yields the proposition.


## 6. Singular Analytic Solutions on Sectors to the Main Problem

We go back to the sequence of formal constructions performed in Section4 under the new light shed in Section 5 on problem (95).We first recall the definitions of a good covering and associated sets of sectors as introduced in [3].Definition 17.
Letς≥2 be an integer. For all 0≤p≤ς-1, we consider open sectors Ep centered at 0, with radius ϵ0>0 and opening π/(χ+α)κ+ξp with ξp>0 small enough such that Ep∩Ep+1≠∅, for all 0≤p≤ς-1 (with the convention that Eς=E0). Moreover, we assume that the intersection of any three different elements in {Ep}0≤p≤ς-1 is empty and that ∪p=0ς-1Ep=U∖{0}, where U is some neighborhood of 0 in C. Such a set of sectors {Ep}0≤p≤ς-1 is called a good covering in C∗.Definition 18.
Let{Ep}0≤p≤ς-1 be a good covering in C∗. Let T be an open bounded sector centered at 0 with radius rT and consider a family of open sectors (161)Sdp,θ,ϵ0rT=T∈C∗∣T<ϵ0rT,dp-arg⁡T<θ2with aperture θ>π/κ and where dp∈R, for all 0≤p≤ς-1, are directions which satisfy the following constraints: let qlm be the roots of the polynomials (100) defined by (101) and Sdp and 0≤p≤ς-1 be unbounded sectors centered at 0 with directions dp and with small aperture. We assume the following.
(1) There exists a constantM1>0 such that(162)τ-qlm≥M11+τfor all 0≤l≤δDκ-1; all m∈R; all τ∈Sdp∪D¯(0,ρ), for all 0≤p≤ς-1.
(2) There exists a constantM2>0 such that(163)τ-ql0m≥M2ql0mfor some l0∈{0,…,δDκ-1}; all m∈R; all τ∈Sdp∪D¯(0,ρ), for all 0≤p≤ς-1.
(3) For all0≤p≤ς-1, all t∈T; all ϵ∈Ep; we have that ϵα+χt∈Sdp,θ,ϵ0α+χrT.
We say that the family{(Sdp,θ,ϵ0rT)0≤p≤ς-1,T} is associated with the good covering {Ep}0≤p≤ς-1.In the next main first outcome, we construct a family of actual holomorphic solutions to the principal equation (61) which may be meromorphic at (ϵ,t)=(0,0) and defined on the sectors Ep with respect to the complex parameter ϵ. Furthermore, we can also control the difference between any two neighboring solutions on the intersections Ep∩Ep+1 and state that it is exponentially flat of order at most (χ+α)κ with respect to ϵ.Theorem 19.
One considers the nonlinear singularly perturbed PDE (61) and takes for granted that all the assumptions (60), (62), (65), (66), (67), (77), (78), (79), (88), (99), (105), (106), and (107) hold for some rational numbers α>1, β∈Q and integers γ∈Z, κ≥1. Let {Ep}0≤p≤ς-1 a good covering in C∗ be given, for which a family of open sectors {(Sdp,θ,ϵ0rT)0≤p≤ς-1,T} associated with this good covering can be singled out.
Then, there exist a radiusrQ~,R~D>0 large enough and ϵ0>0 small enough, for which a family {udp(t,z,ϵ)}0≤p≤ς-1 of actual solutions of (61) can be built up. More exactly, the functions udp(t,z,ϵ) solve the following singularly perturbed PDE:(164)Q~∂z∑l=0qalϵmltkludpt,z,ϵ+∑l=0Mclϵμlthludp2t,z,ϵ=∑j=0pb~jzϵnjtbj+Fϵαt,ϵ+∑l=1DϵΔltdl∂tδlR~l∂zut,z,ϵwith an additional part of forcing term Fϵαt,ϵ where FT,ϵ is given by expression (183) and defines a holomorphic bounded function provided that the additional constraints (184) are fulfilled. Each function udp(t,z,ϵ) can be decomposed as(165)udpt,z,ϵ=ϵβ-a0c0ϵαtk0-h0-a0c0ϵαtk0-h0Jϵαt+ϵαtγvdpt,z,ϵ,where JT is holomorphic on some disc D(0,dJ), dJ>0, and vdp(t,z,ϵ) defines a bounded holomorphic function on T×Hβ′×Ep for any given 0<β′<β, with vdp(0,z,ϵ)≡0 on Hβ′×Ep. Furthermore, there exist constants Kp,Mp>0 and σ>0 (independent of ϵ) such that(166)supt∈T∩D0,σ,z∈Hβ′⁡vdp+1t,z,ϵ-vdpt,z,ϵ≤Kpexp⁡-Mpϵχ+ακfor all ϵ∈Ep+1∩Ep, for all 0≤p≤ς-1 (where by convention vdς=vd0).Proof.
We plan to construct actual solutions of the main equation (61) by performing backwards the sequence of constructions described in Section 4 starting from problem (95) solved in Section 5.
Let{Ep}0≤p≤ς-1 be a good covering in C∗ and let {(Sdp,θ,ϵ0rT)0≤p≤ς-1,T} be a family of sectors associated with this good covering. From Proposition 15, we see that, for each direction dp, one can get a solution ωκdp(τ,m,ϵ) of the convolution equation (95) that belongs to the space F(ν,β,μ,χ,α,κ,ϵ)dp and thus satisfies the following bounds:(167)ωκdpτ,m,ϵ≤ϖ1+m-μe-βmτ/ϵχ+α1+τ/ϵχ+α2κexpντϵχ+ακfor all τ∈D¯(0,ρ)∪Sdp, all m∈R, and all ϵ∈D(0,ϵ0)∖{0}, for some well chosen ϖ>0. Besides, these functions ωκdp(τ,m,ϵ) are analytic continuations with respect to τ of a common convergent series (168)ωκτ,m,ϵ=∑n≥1ωnm,ϵΓn/κτnwith coefficients in the Banach space Eβ,μ solution of (95) for all τ∈D(0,ρ). In particular, we see that the formal power series (169)ΩκT,m,ϵ=∑n≥1ωnm,ϵTnis mκ-summable in direction dp as a series with coefficients in the Banach space E(β,μ) for all ϵ∈D(0,ϵ0)∖{0} in the sense of Definition 7. We denote by(170)ΩκdpT,m,ϵ=κ∫Lγωκdpu,m,ϵexp⁡-uTκduuits mκ-sum in direction dp, where Lγ=R+eiγ⊂Sdp, which defines an Eβ,μ-valued analytic function with respect to T on a sector (171)Sdp,θ,h′ϵχ+α=T∈C∗:T<h′ϵχ+α,dp-arg⁡T<θ2for π/κ<θ<π/κ+ApSdp (where ApSdp denotes the aperture of the sector Sdp) and some h′>0 (independent of ϵ), for all ϵ∈D0,ϵ0∖{0}.
Bearing in mind the identities of Proposition8 and using the properties for the mκ-sum with respect to derivatives and products (within the Banach algebra E=Eβ,μ equipped with the convolution product ⋆ as described in Proposition 2), we check that the functions Ωκdp(T,m,ϵ) must solve the following problem:(172)LT,m,ϵΩκdpT,m,ϵ=RT,m,ϵΩκdpT,m,ϵ,where(173)LT,m,ϵΩκdpT,m,ϵ=Q~imΩκdpT,m,ϵ-a0+∑l=1salϵ-χkl-k0Tkl-k0+∑l=s+1qalϵml+β-αkl-χkl-k0Tkl-k0-2a0c0∑l=1s′clϵ-χhl-h0Thl-h0+∑l=s′+1Mclϵμl+2β-αhl-χhl-h0Thl-h0-2a0c0Jϵ-χT∑l=0s′clϵ-χhl-h0Thl-h0+∑l=s′+1Mclϵμl+2β-αhl-χhl-h0Thl-h0+Q~im12π1/2∫-∞+∞ΩκdpT,m-m1,ϵΩκdpT,m1,ϵdm1ϵ-χ-k0+γT-k0+γ×∑l=0s′clϵ-χhlThl+∑l=s′+1Mclϵμl+2β-αhl-χhlThl,(174)RT,m,ϵΩκdpT,m,ϵ=∑j=0pB~jmϵnj-αbj-χbj-k0-γTbj-k0-γ+∑l=1D-1ϵΔl+αδl-dl+β×∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dϵ-χdl-k0-q1-q2R~lim×Tdl,q1,q2Tκ+1∂Tq2+∑1≤p≤q2-1Aq2,pTκq2-pTκ+1∂TpΩκdpT,m,ϵ+ϵΔD+αδD-dD+β×∑q1+q2=δD,q1≥1δD!q1!q2!∏d=0q1-1γ-dϵ-χdD-k0-q1-q2R~Dim×TdD,q1,q2Tκ+1∂Tq2+∑1≤p≤q2-1Aq2,pTκq2-pTκ+1∂TpΩκdpT,m,ϵ+R~DimTκ+1∂TδD+∑1≤p≤δD-1AδD,pTκδD-pTκ+1∂TpΩκdpT,m,ϵ.We examine now the function(175)VdpT,z,ϵ=F-1m⟼ΩκdpT,m,ϵzwhich defines a bounded holomorphic function with respect to T on Sdp,θ,h′ϵχ+α, with respect to z on Hβ′ for any 0<β′<β, and for all ϵ on D0,ϵ0∖{0}. Using the properties of the Fourier inverse transform described in Proposition 9 and watching out the expansions (91), we extract from equality (172) the next equation satisfied by VdpT,z,ϵ; namely,(176)Q~∂zVdpT,z,ϵ-a0+∑l=1salϵ-χkl-k0Tkl-k0+∑l=s+1qalϵml+β-αkl-χkl-k0Tkl-k0-2a0c0∑l=1s′clϵ-χhl-h0Thl-h0+∑l=s′+1Mclϵμl+2β-αhl-χhl-h0Thl-h0-2a0c0Jϵ-χT∑l=0s′clϵ-χhl-h0Thl-h0+∑l=s′+1Mclϵμl+2β-αhl-χhl-h0Thl-h0+Q~∂zVdp2T,z,ϵϵ-χ-k0+γT-k0+γ∑l=0s′clϵ-χhlThl+∑l=s′+1Mclϵμl+2β-αhl-χhlThl=∑j=0pb~jzϵnj-αbj-χbj-k0-γTbj-k0-γ+∑l=1D-1ϵΔl+αδl-dl+β×∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dϵ-χdl-k0-q1Tdl-k0-q1R~l∂zϵχq2∂Tq2VdpT,z,ϵ+ϵΔD+αδD-dD+β∑q1+q2=δD,q1≥1δD!q1!q2!∏d=0q1-1γ-dϵ-χdD-k0-q1TdD-k0-q1×R~D∂zϵχq2∂Tq2VdpT,z,ϵ+TdD-k0R~D∂z∂TδDVdpT,z,ϵ.We now set T=ϵχT and we focus on the function(177)VdpT,z,ϵ=VdpϵχT,z,ϵwhich defines a bounded holomorphic function with respect to T such that T∈ϵ-χSdp,θ,h′ϵχ+α and with respect to z on Hβ′ for any 0<β′<β, for all ϵ∈D0,ϵ0∖{0}. Having a quick look at (176), we observe that Vdp(T,z,ϵ) solves a related equation which after multiplication by Tk0+γ yields(178)Q~∂zVdpT,z,ϵ-a0Tk0+γ+∑l=1salTkl+∑l=s+1qalϵml+β-αklTklTγ-2a0c0Tk0-h0+γ∑l=1s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl-2a0c0Tk0-h0+γJT∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl+Q~∂zVdp2T,z,ϵT2γ∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl=∑j=0pb~jzϵnj-αbjTbj+∑l=1DϵΔl+αδl-dl+β∑q1+q2=δlδl!q1!q2!∏d=0q1-1γ-dTdl+γ-q1R~l∂z∂Tq2VdpT,z,ϵ.In the next step, we introduce the function(179)UdpT,z,ϵ=-a0c0Tk0-h0-a0c0Tk0-h0JT+TγVdpT,z,ϵwhich defines a holomorphic function with respect to T such that T∈ϵ-χSdp,θ,h′ϵχ+α and with respect to z on Hβ′ for any 0<β′<β, for all ϵ∈D0,ϵ0∖{0}. Notice that this function may be meromorphic at T=0, provided that h0>k0. Taking (178) into consideration, we see that the function UdpT,z,ϵ solves the next PDE with forcing term(180)Q~∂z∑l=0qalϵml+β-αklTklUdpT,z,ϵ+∑l=0Mclϵμl+2β-αhlThlUdp2T,z,ϵ=∑j=0pb~jzϵnj-αbjTbj+FT,ϵ+∑l=1DϵΔl+αδl-dl+βTdlR~l∂z∂TδlUdpT,z,ϵwhich is exactly (72) announced in Remark 12 of Section 4.1, where F(T,ϵ) is a contribution to the forcing term equal to(181)FT,ϵ=-Q~0a0c0Tk0-h0+a0c0Tk0-h0JT×-a0Tk0+∑l=1salTkl+∑l=s+1qalϵml+β-αklTkl-2a0c0Tk0-h0∑l=1s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl-2a0c0Tk0-h0JT∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl-Q~0a0c0Tk0-h0+a0c0Tk0-h0JT2∑l=0s′clThl+∑l=s′+1Mclϵμl+2β-αhlThl+∑l=1DϵΔl+αδl-dl+βTdlR~l0∂Tδla0c0Tk0-h0+a0c0Tk0-h0JT.Using the fact that U0T solves the second-order algebraic equation (70) and noticing the following identity: (182)-a0Tk0+∑l=1salTkl-2a0c0Tk0-h0∑l=1s′clThl=∑l=0salTkl-2a0c0Tk0-h0∑l=0s′clThl,we can abridge the latter expression of FT,ϵ as(183)FT,ϵ=Q~0U0T∑l=s+1qalϵml+β-αklTkl-2a0c0Tk0-h0∑l=s′+1Mclϵμl+2β-αhlThl-2a0c0Tk0-h0JT∑l=s′+1Mclϵμl+2β-αhlThl-Q~0U02T∑l=s′+1Mclϵμl+2β-αhlThl+∑l=1DϵΔl+αδl-dl+βTdlR~l0∂Tδla0c0Tk0-h0+a0c0Tk0-h0JT.Observe that FT,ϵ is bounded holomorphic with respect to ϵ and is analytic in T near 0 provided that the following additional conditions hold:(184)kl+k0-h0≥0,hp+2k0-h0≥0,dm+k0-h0-δm≥0for all s+1≤l≤q, s′+1≤p≤M, and 1≤m≤D.
Finally, we put(185)udpt,z,ϵ=ϵβUdpϵαt,z,ϵ=ϵβU0ϵαt+ϵαtγVdpϵχ+αt,z,ϵwhich defines a holomorphic function with respect to t on T, with respect to z∈Hβ′ for any 0<β′<β, and with respect to ϵ∈Ep, where T and Ep are sectors described in Definition 18. As a result, udpt,z,ϵ admits the decomposition (165) with vdpt,z,ϵ=Vdpϵχ+αt,z,ϵ which determines a bounded holomorphic function on T×Hβ′×Ep for any given 0<β′<β with the property vdp0,z,ϵ≡0 for all z,ϵ∈Hβ′×Ep. Again, the function udpt,z,ϵ may be meromorphic in both t and ϵ in the vicinity of the origin. From (180) and (183) we deduce that udpt,z,ϵ solves the next main problem(186)Q~∂z∑l=0qalϵmltkludpt,z,ϵ+∑l=0Mclϵμlthludp2t,z,ϵ=∑j=0pb~jzϵnjtbj+Fϵαt,ϵ+∑l=1DϵΔltdl∂tδlR~l∂zudpt,z,ϵwith additional forcing term F(ϵαt,ϵ). As a spin-off, by applying the operator ∂zv on the left- and right-hand side of this last equation, we see that udp(t,z,ϵ) is also an actual solution of problem (61) disclosed at the beginning of Section 4.
In the last part of the proof, we proceed to justify estimates (166). The steps of the verification are similar to the arguments displayed in Theorem  1 of [3], but we choose to present them for the sake of completeness. Let p∈{0,…,ς-1}. By the sequence of constructions performed above, we see that the function vdp(t,z,ϵ) can be written as a mκ-Laplace and Fourier transform(187)vdpt,z,ϵ=κ2π1/2∫-∞+∞∫Lγpωκdpu,m,ϵexp⁡-uϵχ+αtκeizmduudm,where Lγp=R+eiγp⊂Sdp. Using the fact that the function u↦ωκu,m,ϵexp(-(u/ϵχ+αt)κ)/u is holomorphic on D(0,ρ) for all (m,ϵ)∈R×(D(0,ϵ0)∖{0}), its integral along the union of a segment starting from 0 to (ρ/2)eiγp+1, an arc of circle with radius ρ/2 which connects (ρ/2)eiγp+1 and (ρ/2)eiγp, and a segment starting from (ρ/2)eiγp to 0 are vanishing. Therefore, we can write the difference vdp+1-vdp as a sum of three integrals:(188)vdp+1t,z,ϵ-vdpt,z,ϵ=κ2π1/2∫-∞+∞∫Lρ/2,γp+1ωκdp+1u,m,ϵe-u/ϵχ+αtκeizmduudm-κ2π1/2∫-∞+∞∫Lρ/2,γpωκdpu,m,ϵe-u/ϵχ+αtκeizmduudm+κ2π1/2∫-∞+∞∫Cρ/2,γp,γp+1ωκu,m,ϵe-u/ϵχ+αtκeizmduudm,where Lρ/2,γp+1=[ρ/2,+∞)eiγp+1, Lρ/2,γp=[ρ/2,+∞)eiγp, and Cρ/2,γp,γp+1 is an arc of circle with radius connecting (ρ/2)eiγp and (ρ/2)eiγp+1 with a well chosen orientation.
We give estimates for the quantity(189)I1=κ2π1/2∫-∞+∞∫Lρ/2,γp+1ωκdp+1u,m,ϵe-u/ϵχ+αtκeizmduudm.By construction, the direction γp+1 (which depends on ϵχ+αt) is chosen in such a way that cos(κ(γp+1-arg(ϵχ+αt)))≥δ1, for all ϵ∈Ep∩Ep+1, for all t∈T, and for some fixed δ1>0. From the estimates (167), we get that(190)I1≤κ2π1/2∫-∞+∞∫ρ/2+∞ϖ1+m-μe-βmr/ϵχ+α1+r/ϵχ+α2κ×exp⁡νrϵχ+ακexp⁡-cosκγp+1-argϵχ+αtϵχ+αtκrκe-mIm⁡zdrrdm≤κϖ2π1/2∫-∞+∞e-β-β′mdm∫ρ/2+∞1ϵχ+αexp⁡-δ1tκ-νrϵχ+ακdr≤2κϖ2π1/2∫0+∞e-β-β′mdm∫ρ/2+∞ϵχ+ακ-1δ1/tκ-νκρ/2κ-1×δ1/tκ-νκrκ-1ϵχ+ακexp⁡-δ1tκ-νrϵχ+ακdr≤2κϖ2π1/2ϵχ+ακ-1β-β′δ1/tκ-νκρ/2κ-1exp⁡-δ1tκ-νρ/2κϵχ+ακ≤2κϖ2π1/2ϵχ+ακ-1β-β′δ2κρ/2κ-1exp⁡-δ2ρ/2κϵχ+ακfor all t∈T and Imz≤β′ with t<(δ1/(δ2+ν))1/κ, for some δ2>0, and for all ϵ∈Ep∩Ep+1.
In the same way, we also give estimates for the integral(191)I2=κ2π1/2∫-∞+∞∫Lρ/2,γpωκdpu,m,ϵe-u/ϵχ+αtκeizmduudm.Namely, the direction γp (which depends on ϵχ+αt) is chosen in such a way that cosκγp-arg⁡ϵχ+αt≥δ1, for all ϵ∈Ep∩Ep+1, for all t∈T, and for some fixed δ1>0. Again from the estimates (167) and following the same steps as in (190), we deduce that(192)I2≤2κϖ2π1/2ϵχ+ακ-1β-β′δ2κρ/2κ-1exp⁡-δ2ρ/2κϵχ+ακfor all t∈T and Imz≤β′ with t<(δ1/(δ2+ν))1/κ, for some δ2>0, and for all ϵ∈Ep∩Ep+1.
Finally, we give upper bound estimates for the integral(193)I3=κ2π1/2∫-∞+∞∫Cρ/2,γp,γp+1ωκu,m,ϵe-u/ϵχ+αtκeizmduudm.By construction, the arc of circle Cρ/2,γp,γp+1 is chosen in such a way that cosκθ-argϵχ+αt≥δ1, for all θ∈[γp,γp+1] (if γp<γp+1) and θ∈[γp+1,γp] (if γp+1<γp), for all t∈T, for all ϵ∈Ep∩Ep+1, and for some fixed δ1>0. Bearing in mind (167) and (118), we get that(194)I3≤κ2π1/2∫-∞+∞∫γpγp+1ϖ1+m-μe-βmρ/2/ϵχ+α1+ρ/2/ϵχ+α2κ×exp⁡νρ/2ϵχ+ακexp⁡-cos⁡κθ-argϵχ+αtϵχ+αtκρ2κe-mIm⁡zdθdm≤κϖ2π1/2∫-∞+∞e-β-β′mdm×γp-γp+1ρ/2ϵχ+αexp⁡-δ1/tκ-ν2ρ/2ϵχ+ακ×exp⁡-δ1/tκ-ν2ρ/2ϵχ+ακ≤2κϖγp-γp+12π1/2β-β′supx≥0⁡x1/κe-δ1/tκ-νx×exp⁡-δ1/tκ-ν2ρ/2ϵχ+ακ≤2κϖγp-γp+12π1/2β-β′1/κδ21/κe-1/κexp⁡-δ22ρ/2ϵχ+ακfor all t∈T and Imz≤β′ with t<(δ1/(δ2+ν))1/κ, for some δ2>0, and for all ϵ∈Ep∩Ep+1.
Finally, gathering the three above inequalities (190), (192), and (194), we deduce from decomposition (188) that (195)vdp+1t,z,ϵ-vdpt,z,ϵ≤4κϖ2π1/2ϵχ+ακ-1β-β′δ2κρ/2κ-1exp⁡-δ2ρ/2κϵχ+ακ+2κϖγp-γp+12π1/2β-β′1/κδ21/κe-1/κexp⁡-δ22ρ/2ϵχ+ακfor all t∈T and Imz≤β′ with t<(δ1/(δ2+ν))1/k, for some δ2>0, and for all ϵ∈Ep∩Ep+1. Therefore, inequality (166) holds.


## 7. Parametric Gevrey Asymptotic Expansions of the Solutions

### 7.1.k-Summable Formal Series and Ramis-Sibuya Theorem

We recall the definition ofk-Borel summability of formal series with coefficients in a Banach space as introduced in [5].Definition 20.
Letk≥1 be an integer. A formal series (196)X^ϵ=∑j=0∞ajϵj∈Fϵwith coefficients in a Banach space (F,·F) is said to be k-summable with respect to ϵ in the direction d∈R if
(i) the existence ofρ∈R+ is ensured such that the following formal series, called formal Borel transform of X^ of order k, (197)BkX^τ=∑j=0∞ajτjΓ1+j/k∈Fτis absolutely convergent for τ<ρ;
(ii) one can select aδ>0 such that the series Bk(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 (198)BkX^τF≤CeKτkfor all τ∈Sd,δ.If the definition above is fulfilled, the vector valued Laplace transform of orderk of Bk(X^)(τ) in the direction d is set as (199)LkdBkX^ϵ=ϵ-k∫LγBkX^ue-u/ϵkkuk-1du,along a half line Lγ=R+eiγ⊂Sd,δ∪{0}, where γ depends on ϵ and is chosen in such a way that cos(k(γ-arg⁡(ϵ)))≥δ1>0, for some fixed δ1, for all ϵ in a sector (200)Sd,θ,R1/k=ϵ∈C∗:ϵ<R1/k,d-argϵ<θ2,where π/k<θ<π/k+2δ and 0<R<δ1/K. The function Lkd(Bk(X^))(ϵ) is called the k-sum of the formal series X^(t) in the direction d. It is bounded and holomorphic on the sector Sd,θ,R1/k and has the formal series X^(ϵ) as Gevrey asymptotic expansion of order 1/k with respect to ϵ on Sd,θ,R1/k. This means that, for all π/k<θ1<θ, there exist C,M>0 such that (201)LkdBkX^ϵ-∑p=0n-1apϵpF≤CMnΓ1+nkϵnfor all n≥1, all ϵ∈Sd,θ1,R1/k.Now, we state a cohomological criterion fork-summability of formal series with coefficients in Banach spaces (see [25], p. 121 or [27], Lemma  XI-2-6) which is known as the Ramis-Sibuya theorem in the literature.Theorem (RS).
Let
(
F
,
·
F
)
be a Banach space over
C
and
{
E
p
}
0
≤
p
≤
ς
-
1
be a good covering in
C
∗
. For all
0
≤
p
≤
ς
-
1
, let
G
p
be a holomorphic function from
E
p
into the Banach space
(
F
,
·
F
)
and let the cocycle
Θ
p
(
ϵ
)
=
G
p
+
1
(
ϵ
)
-
G
p
(
ϵ
)
be a holomorphic function from the sector
Z
p
=
E
p
+
1
∩
E
p
into
F
(with the convention that
E
ς
=
E
0
and
G
ς
=
G
0
). One makes the following assumptions.(
1) The functions
G
p
(
ϵ
)
are bounded as
ϵ
∈
E
p
tends to the origin in
C
, for all
0
≤
p
≤
ς
-
1.(
2) The functions
Θ
p
(
ϵ
)
are exponentially flat of order
k
on
Z
p
, for all
0
≤
p
≤
ς
-
1.This means that there exist constantsCp,Ap>0such that(202)ΘpϵF≤Cpe-Ap/ϵkfor allϵ∈Zp, all0≤p≤ς-1.Then, for all
0
≤
p
≤
ς
-
1
, the functions
G
p
(
ϵ
)
are the
k-sums onEpof a commonk-summable formal seriesG^(ϵ)∈F[[ϵ]].


### 7.2. Parametric Gevrey Asymptotic Expansions of the Solutions and Construction of(χ+α)κ-Sums

In this subsection, we state the second main result of our work, namely, the existence of a formal power series in the parameterϵ whose coefficients are bounded holomorphic functions on the product of a sector with small radius centered at 0 and a strip in C2 which is the common Gevrey asymptotic expansion of order 1/(χ+α)κ of the functions vdp(t,z,ϵ) appearing in the expansion (165) of the solutions udp(t,z,ϵ) to the main equations (61) and (164) established in Theorem 19.Theorem 21.
Let one assume that the hypotheses of Theorem19 hold. Then, there exists a formal power series (203)v^t,z,ϵ=∑m≥0vmt,zϵmwhose coefficients vm(t,z) belong to the Banach space F of bounded holomorphic functions on (T∩D(0,σ))×Hβ′ equipped with supremum norm, where σ>0 is defined in Theorem 19, and such that the functions vdp(t,z,ϵ) from the decomposition (165) are its (χ+α)κ-sums on the sectors Ep, for all 0≤p≤ς-1, viewed as holomorphic functions from Ep into F. In other words, for all 0≤p≤ς-1, there exist two constants Cp,Mp>0 such that(204)supt∈T∩D0,σ,z∈Hβ′⁡vdpt,z,ϵ-∑m=0n-1vmt,zϵm≤CpMpnΓ1+nχ+ακϵnfor all n≥1; all ϵ∈Ep.Proof.
We consider the family of functionsvdp(t,z,ϵ), 0≤p≤ς-1, constructed in Theorem 19. For all 0≤p≤ς-1, we define Gp(ϵ)≔(t,z)↦vdp(t,z,ϵ), which is by construction a holomorphic and bounded function from Ep into the Banach space F of bounded holomorphic functions on (T∩D(0,σ))×Hβ′ equipped with the supremum norm, where T is introduced in Definition 18, σ>0 is set in Theorem 19, and β′>0 is the width of the strip Hβ′ on which the coefficients bj(z) are defined with respect to z (see (62)). Bearing in mind the estimates (166), we see that the cocycle Θp(ϵ)=Gp+1(ϵ)-Gp(ϵ) is exponentially flat of order (χ+α)κ on Zp=Ep∩Ep+1, for any 0≤p≤ς-1. Therefore, according to Theorem (RS) stated above, we obtain a formal power series (205)G^ϵ=∑m≥0vmt,zϵm≕v^t,z,ϵ∈Fϵsuch that the functions Gp(ϵ) are the (χ+α)κ-sums on Ep of G^(ϵ) as F-valued functions, for all 0≤p≤ς-1. The result follows.


## 7.1.k-Summable Formal Series and Ramis-Sibuya Theorem

We recall the definition ofk-Borel summability of formal series with coefficients in a Banach space as introduced in [5].Definition 20.
Letk≥1 be an integer. A formal series (196)X^ϵ=∑j=0∞ajϵj∈Fϵwith coefficients in a Banach space (F,·F) is said to be k-summable with respect to ϵ in the direction d∈R if
(i) the existence ofρ∈R+ is ensured such that the following formal series, called formal Borel transform of X^ of order k, (197)BkX^τ=∑j=0∞ajτjΓ1+j/k∈Fτis absolutely convergent for τ<ρ;
(ii) one can select aδ>0 such that the series Bk(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 (198)BkX^τF≤CeKτkfor all τ∈Sd,δ.If the definition above is fulfilled, the vector valued Laplace transform of orderk of Bk(X^)(τ) in the direction d is set as (199)LkdBkX^ϵ=ϵ-k∫LγBkX^ue-u/ϵkkuk-1du,along a half line Lγ=R+eiγ⊂Sd,δ∪{0}, where γ depends on ϵ and is chosen in such a way that cos(k(γ-arg⁡(ϵ)))≥δ1>0, for some fixed δ1, for all ϵ in a sector (200)Sd,θ,R1/k=ϵ∈C∗:ϵ<R1/k,d-argϵ<θ2,where π/k<θ<π/k+2δ and 0<R<δ1/K. The function Lkd(Bk(X^))(ϵ) is called the k-sum of the formal series X^(t) in the direction d. It is bounded and holomorphic on the sector Sd,θ,R1/k and has the formal series X^(ϵ) as Gevrey asymptotic expansion of order 1/k with respect to ϵ on Sd,θ,R1/k. This means that, for all π/k<θ1<θ, there exist C,M>0 such that (201)LkdBkX^ϵ-∑p=0n-1apϵpF≤CMnΓ1+nkϵnfor all n≥1, all ϵ∈Sd,θ1,R1/k.Now, we state a cohomological criterion fork-summability of formal series with coefficients in Banach spaces (see [25], p. 121 or [27], Lemma  XI-2-6) which is known as the Ramis-Sibuya theorem in the literature.Theorem (RS).
Let
(
F
,
·
F
)
be a Banach space over
C
and
{
E
p
}
0
≤
p
≤
ς
-
1
be a good covering in
C
∗
. For all
0
≤
p
≤
ς
-
1
, let
G
p
be a holomorphic function from
E
p
into the Banach space
(
F
,
·
F
)
and let the cocycle
Θ
p
(
ϵ
)
=
G
p
+
1
(
ϵ
)
-
G
p
(
ϵ
)
be a holomorphic function from the sector
Z
p
=
E
p
+
1
∩
E
p
into
F
(with the convention that
E
ς
=
E
0
and
G
ς
=
G
0
). One makes the following assumptions.(
1) The functions
G
p
(
ϵ
)
are bounded as
ϵ
∈
E
p
tends to the origin in
C
, for all
0
≤
p
≤
ς
-
1.(
2) The functions
Θ
p
(
ϵ
)
are exponentially flat of order
k
on
Z
p
, for all
0
≤
p
≤
ς
-
1.This means that there exist constantsCp,Ap>0such that(202)ΘpϵF≤Cpe-Ap/ϵkfor allϵ∈Zp, all0≤p≤ς-1.Then, for all
0
≤
p
≤
ς
-
1
, the functions
G
p
(
ϵ
)
are the
k-sums onEpof a commonk-summable formal seriesG^(ϵ)∈F[[ϵ]].


## 7.2. Parametric Gevrey Asymptotic Expansions of the Solutions and Construction of(χ+α)κ-Sums

In this subsection, we state the second main result of our work, namely, the existence of a formal power series in the parameterϵ whose coefficients are bounded holomorphic functions on the product of a sector with small radius centered at 0 and a strip in C2 which is the common Gevrey asymptotic expansion of order 1/(χ+α)κ of the functions vdp(t,z,ϵ) appearing in the expansion (165) of the solutions udp(t,z,ϵ) to the main equations (61) and (164) established in Theorem 19.Theorem 21.
Let one assume that the hypotheses of Theorem19 hold. Then, there exists a formal power series (203)v^t,z,ϵ=∑m≥0vmt,zϵmwhose coefficients vm(t,z) belong to the Banach space F of bounded holomorphic functions on (T∩D(0,σ))×Hβ′ equipped with supremum norm, where σ>0 is defined in Theorem 19, and such that the functions vdp(t,z,ϵ) from the decomposition (165) are its (χ+α)κ-sums on the sectors Ep, for all 0≤p≤ς-1, viewed as holomorphic functions from Ep into F. In other words, for all 0≤p≤ς-1, there exist two constants Cp,Mp>0 such that(204)supt∈T∩D0,σ,z∈Hβ′⁡vdpt,z,ϵ-∑m=0n-1vmt,zϵm≤CpMpnΓ1+nχ+ακϵnfor all n≥1; all ϵ∈Ep.Proof.
We consider the family of functionsvdp(t,z,ϵ), 0≤p≤ς-1, constructed in Theorem 19. For all 0≤p≤ς-1, we define Gp(ϵ)≔(t,z)↦vdp(t,z,ϵ), which is by construction a holomorphic and bounded function from Ep into the Banach space F of bounded holomorphic functions on (T∩D(0,σ))×Hβ′ equipped with the supremum norm, where T is introduced in Definition 18, σ>0 is set in Theorem 19, and β′>0 is the width of the strip Hβ′ on which the coefficients bj(z) are defined with respect to z (see (62)). Bearing in mind the estimates (166), we see that the cocycle Θp(ϵ)=Gp+1(ϵ)-Gp(ϵ) is exponentially flat of order (χ+α)κ on Zp=Ep∩Ep+1, for any 0≤p≤ς-1. Therefore, according to Theorem (RS) stated above, we obtain a formal power series (205)G^ϵ=∑m≥0vmt,zϵm≕v^t,z,ϵ∈Fϵsuch that the functions Gp(ϵ) are the (χ+α)κ-sums on Ep of G^(ϵ) as F-valued functions, for all 0≤p≤ς-1. The result follows.



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*Source: 9405298-2017-09-13.xml*