# On Initial-Boundary Value Problems for Hyperbolic Equations in Domains with Conical Points

Authors: Nguyen Manh Hung; Nguyen Thanh Anh
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider an initial-boundary value problem for general higher-order hyperbolic equation in an infinite cylinder with the base containing conical points on the boundary. We establish several results on the unique solvability, the regularity, and the asymptotic behaviour of the solution near the conical points.


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## Body

## 1. Introduction

A large number of investigations have been devoted to boundary value problems in nonsmooth domains with conical points. Up to now, elliptic boundary value problems in domains with point singularities have been thoroughly investigated (see, e.g., [1–3]).We are concerned with hyperbolic equations in domains with conical points. This topic has been investigated in many works with different approaches. For example, [4, 5] the Cauchy-Dirichlet and Cauchy-Neumann problems for second-order hyperbolic systems with the coefficients independent of the time variable were treated in which the asymptotics of the solutions were established with explicit formulas for the coefficients. In [6–9], initial-boundary value problems for general higher-order hyperbolic equations and systems with the coefficients depending on both spatial and time variable in a domain containing conical points were studied in which the unique solvability, the regularity, and the asymptotic behaviour of the solutions near the conical points were obtained. In the present paper, these results are extended to initial-boundary value problems for general higher-order hyperbolic equations with more general boundary conditions in infinite cylinders with the bases containing conical points. Such boundary conditions have been considered for elliptic equations in [10, 11] and for parabolic equations in [12, 13].Our paper is organized as follows. Section2 is devoted to some notations and the formulation of the problem. The main results will be stated in Section 3. The proofs of the main theorems will be given in Sections 4 and 5.


## 2. Notations and the Formulation of the Problem

LetG be a bounded domain in ℝn(n⩾2) with the boundary ∂G. We suppose that Γ=∂G∖{0} is a smooth manifold and G in a neighborhood U of the origin 0 coincides with the cone K={x:x/|x|∈Ω}, where Ω is a smooth domain on the unit sphere Sn-1 in ℝn. For each T, 0<T⩽∞, denote QT=G×(0,T),  GT=G×{T}. Specifically, we set Q=Q∞=G×ℝ+*,S=Γ×ℝ+, where ℝ+*=(0,∞), ℝ+=[0,∞). For each multiindex α=(α1,…,αn)∈ℕn, set |α|=α1+⋯+αn and ∂xα=∂xα=∂x1α1⋯∂xnαn. For a nonnegative integer k we write utk instead of ∂tku.We introduce the following differential operator:(1)Lu=L(x,t,∂)u=∑|α|,|β|⩽m‍(-1)|α|∂α(aαβ∂βu),
where aαβ=aαβ(x,t) are bounded complex-valued functions defined in Q¯. We assume that aαβ=(-1)|α|+|β|a¯βα for all |α|,|β|⩽m. This means the differential operator L is formally self-adjoint. We assume further that there exists a positive constant μ0 such that
(2)∑|α|=|β|=m‍aαβ(x,t)ηβηα¯≥μ0∑|α|=m‍|ηα|2,
for all ηα∈ℂ, |α|=m and all (x,t)∈Q¯.We introduce also a system of boundary operators:(3)Bj=Bj(x,t,∂x)=∑|α|⩽μj‍bj,α(x,t)∂xα,j=1,…,m,
on S with smooth coefficients in Q¯. Suppose that
(4)ordBj=μj⩽m-1forj=1,…,χ,m⩽ordBj=μj⩽2m-1forj=χ+1,…,m,
and the coefficients of Bj are independent of t if ordBj<m, where ordBj stands for the order of the differential operator Bj. Suppose that  {Bj(x,t,∂x)}j=1m is a normal system on S for all t∈ℝ+; that is, the two following conditions are satisfied:(i)
μj≠μk for j≠k,(ii)
Bj∘(x,t,ν(x))≠0 for all (x,t)∈S,j=1,…,m.Here ν(x) is the unit outer normal to S at point x, and Bj∘(x,t,∂x) is the principal part of Bj(x,t,∂x),
(5)Bj∘=Bj∘(x,t,∂x)=∑|α|=μj‍bj,α(x,t)∂xα,j=1,…,m.
Furthermore, we assume that Bj∘(0,t,ν(x))≠0 for all x∈S sufficiently close to the origin 0.To be able to reduce the problem considered to variational form we assume that there are boundary operatorsΦj on S, j=1,…,m, such that
(6)B(t,u,v)=∫G‍Luv¯dx+∑j=1χ‍∫S‍ΦjuBjv¯ds+∑j=χ+1m‍∫S‍BjuΦjv¯ds,
for all u,v∈C0∞(G¯∖{0}) and a.e. t∈ℝ+. Here
(7)B(t,u,v)=∑|α|,|β|=0m‍∫G‍aαβ(·,t)∂xβu∂xαv¯dx,t∈ℝ+,
is the bilinear form associated with the operator L(x,t,∂x). Of course this is an essential restriction on the structure of the boundary operators in (3). However, if the system of boundary operators in (3) is a Dirichlet system (then all ord of Bj are less than m) or a generalized Neumann system (then m⩽ordBj⩽2m-1 for all j=1,…,m), the equality (6) holds for a suitable system Φj,j=1,…,m (see [10, Section I.7.]).In this paper, we consider the following problem:(8)utt+Lu=finQ,Bju=0onS,j=1,…,m,(9)u∣t=0=ut∣t=0=0inG.Before giving the definition of generalized solutions to this problem, let us introduce some needed functional spaces.Let us denote byHl(G)(l∈ℕ) the usual Sobolev space of all functions u defined in G with the norm
(10)∥u∥Hl(G)=(∫G‍∑|α|⩽l‍|Dαu|2dx)1/2<∞.
If l⩾1, by Hl-(1/2)(Γ) we denote the space of traces of functions in Hl(G) on Γ with the norm
(11)∥u∥Hl-(1/2)(Γ)=inf{∥v∥Hl(G):v∈Hl(G),v∣Γ=u}.
We set
(12)HBm(G)={u∈Hm(G):Bju=0onSforj=1,…,χ},
with the same norm as in Hm(G). By V2,αl(G)(α∈ℝ) we denote the closure of C0∞(G¯∖{0}) with respect to the norm
(13)∥u∥V2,αl(G)=(∑|p|⩽l‍∫G‍r2(α+|p|-l)|Dpu|2dx)1/2,
where r=|x|=(∑k=1n‍xk2)1/2. By Hαl(G) we denote the weighted Sobolev space of functions u defined in G with the norm
(14)∥u∥Hαl(G)=(∑|p|⩽l‍∫G‍r2α|Dpu|2dx)1/2<+∞.
If l⩾1, then Vαl-1/2(Γ),Hαl-1/2(Γ) denote the spaces consisting of traces of functions from respective spaces V2,αl(G),Hαl(G) on the boundary Γ with the respective norms
(15)∥u∥Vαl-1/2(Γ)=inf{∥v∥V2,αl(G):v∈V2,αl(G),v∣Γ=u},∥u∥Hαl-1/2(Γ)=inf{∥v∥Hαl(G):v∈Hαl(G),v∣Γ=u}.LetX,Y be Banach spaces, 0<T⩽∞. We denote by L2((0,T),X) the space of all functions u:(0,T)→X such that
(16)∥u∥L2((0,T),X)=(∫0T‍∥u(t)∥X2dt)1/2<∞,
and by H1((0,T),X,Y) the space of all functions u∈L2((0,T),X) such that ut∈L2((0,T),Y). The norm in H1((0,T),X,Y) is defined by
(17)∥u∥H1((0,T),X,Y)=(∥u∥L2((0,T),X)2+∥ut∥L2((0,T),Y)2)1/2.
For some γ∈ℝ, we denote by L2(ℝ+*,X,γ) the space of all functions u:ℝ+*→X such that
(18)∥u∥L2(ℝ+*,X,γ)=(∫0∞‍∥u(t)∥X2e-2γtdt)1/2<∞,
and by H1(ℝ+*,X,Y,γ) the space of all functions u∈L2(ℝ+*,X,γ) such that ut∈L2(ℝ+*,Y,γ) with the norm
(19)∥u∥H1(ℝ+*,X,Y,γ)=(∥u∥L2(ℝ+*,X,γ)2+∥ut∥L2(ℝ+*,Y,γ)2)1/2.
For shortness, we set
(20)HBm,1(QT)=H1((0,T),HBm(G),L2(G)),L2(Q,γ)=L2(ℝ+*,L2(G),γ),Hl,0(Q,γ)=L2(ℝ+*,Hl(G),γ),Hl,1(Q,γ)=H1(ℝ+*,Hl(G),L2(G),γ),V2,βl,0(Q,γ)=L2(ℝ+*,V2,βl(G),γ),Hβl,0(Q,γ)=L2(ℝ+*,Hβl(G),γ),Hβl,1(Q,γ)=H1(ℝ+*,Hβl(G),L2(G),γ).Now the definition of generalized solutions of the problem (8)-(9) is given as follows.Definition 1.
Letf be a given function defined on Q which belongs to L2(QT) for each T>0. A function u∈HBm,1(Q,γ) for some γ∈ℝ is called a generalized solution of the problem  (8)-(9) if and only if u∣t=0=0, and for each T>0 the equality
(21)-∫0T‍∫G‍utη-tdt+∑|α|,|β|=0m‍∫0T‍∫G‍aαβ(x,t)∂xβu∂xαη¯dxdt=∫0T‍∫G‍fη-dt,
holds for all η∈HBm,1(QT) satisfying η(·,T)=0.


## 3. Statements of the Main Results

The unique solvability of the problem is given by the following theorem.Theorem 2.
There exists a positive real numberγ0 such that for each γ>γ0, if f∈L2(Q,σ) for some real number σ, the problem (8)-(9) has a unique generalized solution u in the space HBm,1(Q,γ+σ) and
(22)∥u∥HBm,1(Q,γ+σ)2≤C∥f∥L2(Q,σ)2,
where C is a constant independent of u and f.The following theorem states on the regularity of the generalized solution in weighted Sobolev spaces.Theorem 3.
Leth be a positive integer and γ0 be the number as in Theorem 2. Suppose that the function f satisfies the following conditions for some real number σ:(i)
ftk∈L2(Q,kγ0+σ),k⩽h,(ii)
ftk(x,0)=0,0≤k≤h-1.Let u∈HBm,1(Q,γ+σ) for some γ>γ0 be the generalized solution of the problem (8)-(9). Then utk∈Hm2m,1(Q,(k+2)γ+σ) for k=0,1,…,h-1 and
(23)∑k=0h-1‍∥utk∥Hm2m,1(Q,(k+2)γ+σ)2⩽C∑k=0h‍∥ftk∥L2(Q,kγ0+σ)2,
where C is a constant independent of u and f.The proofs of these theorems will be given in Section4. The number γ0 will be defined by formula (66). It is natural that this number should be chosen as small as possible.The remainder of this section is devoted to construct the theorem on the asymptotic behaviour of the solution near the conical point.Letr=|x| and ω be an arbitrary local coordinate system on Sn-1. Let δ be a positive real constant. A differential operator
(24)P(x,t,∂x)=∑|α|⩽l‍pα(x,t)∂xα,
is called δ-admissible operator of order l near the conical point 0 if the coefficients pα(x,t) are infinitely differentiable in Q¯ and there is representation in a neighborhood U of the conical point 0:
(25)pα(x,t)=r|α|-lpα(0)(ω,t)+r|α|-l+δpα(1)(ω,r,t),
where pα(0)∈C∞(Ω¯×ℝ+) and the functions pα(1) are infinitely differentiable functions in Ω¯×ℝ+×ℝ+ such that
(26)|(r∂r)j∂ωβ∂tkpα(1)(ω,r,t)|<cjkβ,
for every multiindex β and every pair of nonnegative integers j,k. Here the constants cjkβ do not depend on ω,r and t. The leading part 𝒫 of the operator P at the point 0 is defined by
(27)𝒫(x,t,∂x)=∑|α|⩽l‍r|α|-lpα(0)(ω,t)∂xα.
It can be directly verified that the derivative ∂xα has the form
(28)∂xα=r-|α|∑p=0|α|‍pαj(ω,∂ω)(r∂r)j,
where pαj(ω,∂ω) are differential operators of order ⩽|α|-j with smooth coefficients on Ω¯. Thus the operator 𝒫(x,t,∂x) can be represented as
(29)𝒫(x,t,∂x)=r-l∑j=0l‍pj(ω,t,∂ω)(r∂r)j.For convenience we rewrite the operatorL(x,t,∂x) in the form
(30)L=L(x,t,∂x)=∑|α|⩽2m‍aα(x,t)∂xα.
Let ℒ=ℒ(x,t,∂x),ℬj=ℬj(x,t,∂x)(j=1,…,m) be the leading parts of L(x,t,∂x), Bj(x,t,∂x) at the point x=0. Since the coefficients of the operators L(x,t,∂x) and Bj(x,t,∂x) are smooth, it is verified easily that
(31)ℒ(x,t,∂x)=∑|α|=2m‍aα(0,t)∂xα,ℬj(x,t,∂x)=∑|α|=μj‍bj,α(0,t)∂xα,
and the operators L(x,t,∂x) and Bj(x,t,∂x) are 1-admissible. Rewrite ℒ(x,t,∂x),ℬj(x,t,∂x) in the form
(32)ℒ(x,t,∂x)=r-2mL(ω,t,∂ω,r∂r),ℬj(x,t,∂x)=r-μjBj(ω,t,∂ω,r∂r).
We introduce the operator
(33)U(λ,t)=(L(ω,t,∂ω,λ),B1(ω,t,∂ω,λ),…,Bm(ω,t,∂ω,λ)),
(λ∈ℂ,t∈ℝ+) of the parameter-depending elliptic boundary value problem
(34)L(ω,t,∂ω,λ)u=finΩ,Bj(ω,t,∂ω,λ)u=gjon∂Ω,j=1,…,m.
This is a pencil of continuous operators from
(35)𝒳≡W2l(Ω)into𝓎≡W2l-2m(Ω)×∏j=1mW2l-μj-(1/2)(∂Ω)(l⩾2m),
depending polynomially on λ∈ℂ.We mention now some well-known definitions [3]. Let t0∈ℝ+ fixed. If λ0∈ℂ, φ0∈𝒳 such that φ0≠0,U(λ0,t0)φ0=0, then λ0 is called an eigenvalue of U(λ,t0) and φ0∈𝒳 is called an eigenvector corresponding to λ0. Λ=dimkerU(λ0,t0) is called the geometric multiplicity of the eigenvalue λ0.If the elementsφ1,…,φs of 𝒳 satisfy the equations
(36)∑q=0σ‍1q!∂βdλqU(λ,t0)∣λ=λ0φσ-q=0forσ=1,…,s,
then the ordered collection φ0,φ1,…,φs is said to be a Jordan chain corresponding to the eigenvalue λ0 of the length s+1. The rank of the eigenvector φ0 (rankφ0) is the maximal length of the Jordan chains corresponding to the eigenvector φ0.A canonical system of eigenvectors ofU(λ0,t0) corresponding to the eigenvalue λ0 is a system of eigenvectors φ1,0,…,φΛ,0 such that rankφ1,0 is maximal among the rank of all eigenvectors corresponding to λ0 and rankφj,0 is maximal among the rank of all eigenvectors in any direct complement in kerU(λ0,t0) to the linear span of the vectors φ1,0,…,φj-1,0(j=2,…,Λ). The numbers κj=rankφj,0(j=1,…,Λ) are called the partial multiplicities and the sum κ=κ1+⋯+κΛ is called the algebraic multiplicity of the eigenvalue λ0.The eigenvalue ofλ0 is called simple if its algebraic multiplicity is equal to one.For each fixedt∈ℝ+ the set of all complex number λ such that U(λ,t) is not invertible is called the spectrum of U(λ,t). It is known that the spectrum of U(λ,t) is an enumerable set of its eigenvalues (see [3, Theorem 5.2.1]). Moreover, there are constants δ,R such that U(λ,t) is invertible for all t∈ℝ+ and all λ in the set
(37)𝒟:={λ∈ℂ:|Reλ|⩽δ|Imλ|,|λ|⩾R},
(see [3, Theorem 3.6.1]).To receive asymptotic formulas of the solutions with the coefficients regular with respect to the variablet we require later that eigenvalues and eigenvectors of the pencil U(λ,t) satisfy the following assumption.Letl1,l2 be nonnegative integers, and let β1,β2 be real numbers such that l1-β1<l2-β2. We say that the assumption (H) for numbers l1,l2,β1,β2 is fulfilled if the following conditions are satisfied.(i)
The linesReλ=-βi+li-n/2(i=1,2) do not contain eigenvalues of the pencil U(λ,t), and the strip -β1+l1-n/2<Reλ<-β2+l2-n/2 contains the eigenvalues λμ(t),μ=1,…,N, with the geometric multiplicities Λμ and the partial multiplicities κμ,k, μ=1,…,N,k=1,…,Λμ, not depending on t∈ℝ+. These eigenvalues are smooth functions on ℝ+.(ii)
A canonical system(38)φk,s(μ)(ω,t),k=1,…,Λμ,s=1,…,κμ,k,
of Jordan chains of U(λ,t) corresponding to the eigenvalue λμ(t) (μ∈{1,…,N}) can be chosen, which consists of functions that are smooth for t∈ℝ+ for all ω∈Ω¯.Theorem 4.
Leth be a positive integer. Let ϵ,β be real numbers such that 0⩽ϵ<1/2, 0<β⩽m and ϵ+n/2, β+n/2 are not integers in the case m⩾n/2. Suppose that the assumptions of Theorem 3 and the assumption (H) for l1=l2=2m, β1=m+ϵ, β2=β are fulfilled. Let u∈Hm,1(Q,γ)(γ>γ0) be the generalized solution of the problem (8)-(9). Suppose further that the operators L(x,t,∂x) and Bj(x,t,∂x) are (m+ϵ-β)- admissible near the origin 0. Then,(i)
for the casem<n/2, the solution u admits the decomposition
(39)u(x,t)=∑μ=1N‍∑k=1Λμ‍∑s=0κμ,k-1‍cμ,k,s(t)rλμ(t)×∑σ=0s‍1σ!(lnr)σφk,s(μ)(ω,t)+w(x,t),
where w, cμ,k,s are functions satisfying wtp∈V2,β2m,0(Q,(p+2)γ+σ), (cμ,k,s)tp∈L2(ℝ+*,(p+2)γ+σ) for p=0,…,h,(ii)
for the casem⩾n/2, we assume further that if λk(t)=q is integer for some t∈ℝ+ and some k,1⩽k⩽N, then λk(t)=q for all t∈ℝ+; then the solution u admits the decomposition
(40)u(x,t)=∑|α|⩽2m-ℓ-1‍cα(t)xα+∑k=2m-ℓ2m-ℓ1-1‍rk∑s=0κk‍1s!(lnr)sφk,s(ω,t)+∑μ=1N‍∑k=1Λμ‍∑s=0κμ,k-1‍cμ,k,s(t)rλμ(t)×∑σ=0s‍1σ!(lnr)σφk,s(μ)(ω,t)+w(x,t),
where w,cα,cμ,k,s are functions satisfying wtp∈V2,β2m,0(Q,(p+2)γ+σ), (cα)tp,(cμ,k,s)tp∈L2(ℝ+*,(p+2)γ+σ), for p=0,…,h, and ℓ1 is the integer with ℓ1-n/2<β<ℓ1+1-n/2, κk=0 if k is not an eigenvalue of U(λ,t); otherwise κk is the maximal partial multiplicity of the eigenvalue k.


## 4. Proofs of Theorems2 and 3

First, let us introduce some more notations. For functionsu,v defined in G and k∈ℕ, we set
(41)|u|k,G=(∫G‍∑|α|=k‍|∂αu|2dx)1/2,(u,v)G=∫G‍uv¯dx.
For functions u and v defined in Q, we set
(42)|u|k,Qτ=(∫0τ‍|u(t)|k,G2dt)1/2,Bt(t,u,v)=∑|α|,|β|⩽m‍∫G‍∂aαβ∂t(t)∂βu(t)∂αv(t)¯dx,Btτ(u,v)=∫0τ‍Bt(t,u,v)dt.
Here and hereafter, we use u(t) instead of u(·,t) for the shortness.To prove Theorem2, it is needed to introduce the Gronwall-Bellman and interpolation inequalities stated in the following lemmas.Lemma 5 (see [14, Lemma 3.1]).
Assumeu,α,β are real-valued continuous on an interval [a,b], β is nonnegative and integrable on [a,b] and α is nondecreasing satisfying
(43)u(τ)⩽α(τ)+∫aτ‍β(t)u(t)dt∀a⩽τ⩽b.
Then
(44)u(τ)⩽α(τ)exp(∫aτ‍β(t)dt)∀a⩽τ⩽b.Lemma 6 (see [15] Theorem4.14).
For each positive real numberϵ and each integer j, 0<j<m, there exists a positive real number C=C(G,m,ϵ) which is dependent only on G, m, and ϵ such that the inequality
(45)|u|j,G2⩽ϵ|u|m,G2+C|u|0,G2,
holds for all u∈Hm(G).Proof of Theorem2.
The theorem is proved by repeating almost word for word the proof of Theorem 3.3 of [7]. Here we present the proof of the existence to show that the restriction of negativeness of σ in that theorem can be omitted.
Let{φk}k=1∞ be a basis of HBm(G) which is orthonormal in L2(G). Put
(46)uN(x,t)=∑k=1N‍ckN(t)φk(x),
where (ckN(t))k=1N are the solution of the system of the following ordinary differential equations of second order:
(47)(uttN(t),φl)G+B(t,uN,φl)=(f(t),φl)G,l=1,…,N,
with the initial conditions
(48)ckN(0)=dckNdt(0)=0,k=1,…,N.
Let us multiply (47) by (d/dt)ckN(t). Take the sum with respect to l from 1 to N to receive
(49)(uttN(t),utN(t))G+B(τ,uN,utN)=(f(t),utN(t))G.
Now adding this equality to its complex conjugate with noting that
(50)B(τ,u,v)¯=B(τ,v,u),
by the formally self-adjointness of the operator L, then integrating the obtained equality with respect to t from 0 to τ with using the integration by parts and (48), we arrive at
(51)|utN(τ)|0,G2+B(τ,uN,uN)=Btτ(uN,uN)+2Re(f,utN)Qτ.
Noting that
(52)|uN(τ)|0,G2=2Re(uN,utN)Qτ,
we have from (51) that
(53)|utN(τ)|0,G2+B0(τ,uN,uN)+ρ|u(τ)|0,G2=Btτ(uN,uN)+2ρRe(uN,utN)Qτ-∑|α|,|β|⩽m|α|+|β|<2m-1‍∫G‍aαβ(τ)∂βuN(τ)∂αuN¯(τ)dx+2Re(f,utN)Qτ.
Now we give estimations for the terms of (53). Firstly, by (2), we see that the left-hand side of (53) is greater than
(54)|utN(τ)|0,G2+μ0|uN(τ)|m,G2+ρ|u(τ)|0,G2.
We write Btτ as the sum of the two following terms:
(55)I1=∑|α|=|β|=m‍∫Qτ‍∂aαβ∂t∂βuN∂αuN¯dxdt,I2=∑|α|,|β|⩽m|α|+|β|⩽2m-1‍∫Qτ‍∂aαβ∂t∂βuN∂αuN¯dxdt.
Put
(56)μ1=sup{|∂aαβ∂t(x,t)|:|α|=|β|=m,(x,t)∈Q¯},m′=∑|α|=m‍1.
Then, by the Cauchy inequality, we have
(57)|I1|⩽μ1∑|α|=|β|=m‍12(|∂βuN|0,Qτ2+|∂αuN|0,Qτ2)⩽m′μ1|uN|m,Qτ2.
By the Cauchy inequality and the interpolation inequality (45), for an arbitrary positive number ϵ1, we have
(58)|I2|⩽ϵ1|uN|m,Qτ2+C1|uN|0,Qτ2,
where C1=C1(ϵ1) is a nonnegative constant independent of uN, f, and τ. Now using again the Cauchy and interpolation inequalities, for an arbitrary positive number ϵ2 with ϵ2<μ0, it holds that
(59)|∑|α|,|β|⩽m|α|+|β|<2m-1‍∫G‍aαβ∂βuN∂αuN¯dx|⩽ϵ2|uN(τ)|m,G2+C2|uN(τ)|0,G2,
where C2=C2(ϵ2) is a nonnegative constant independent of uN, f, and τ. Also by Cauchy inequality we have
(60)|2ρRe(uN,utN)Qτ|⩽(μ0-ϵ2)ρ2m′μ1+ϵ1|uN|0,Qτ2+m′μ1+ϵ1μ0-ϵ2|utN|0,Qτ2.
Now, to deal with the last term of (53), let us consider first the case σ⩾0. In this case, we use the following inequality:
(61)|2Re(f,utN)Qτ|⩽ϵ3|utN|0,Qτ2+1ϵ3|f|0,Qτ2,
where ϵ3>0 arbitrary. Combining the above estimations we get from (53) that
(62)|utN(τ)|0,G2+(μ0-ϵ2)|uN(τ)|m,G2+(ρ-C2)|uN(τ)|0,G2⩽(m′μ1+ϵ1)|uN|m,Qτ2+(C1+(μ0-ϵ2)ρ2m′μ1+ϵ1)|uN|0,Qτ2+(m′μ1+ϵ1μ0-ϵ2+ϵ3)|utN|0,Qτ2+1ϵ3|f|0,Qτ2.
Now fix ϵ1, ϵ2 and consider the function
(63)g(ρ)=C1+(μ0-ϵ2)ρ2/(m′μ1+ϵ1)ρ-C2forρ>C2.
We see that
(64)dgdρ=Aρ2-2AC2ρ-C1(ρ-C2)2withA=(μ0-ϵ2)m′μ1+ϵ1.
We see that the function g has a unique minimum at
(65)ρ0=ρ0(ϵ1,ϵ2)=C2+C22+C1A>C2.
Let us denote
(66)γ0=12infϵ1>00<ϵ2<μ0max{m′μ1+ϵ1μ0-ϵ2,g(ρ0)}.
Now we take real numbers γ, γ1 arbitrarily satisfying γ0<γ1<γ. Then there are positive real numbers ϵ1, ϵ2 (ϵ2<μ0), ρ (ρ>C2(ϵ1,ϵ2)), and ϵ3 such that
(67)m′μ1+ϵ1μ0-ϵ2+ϵ3<2γ1,C1(ϵ1,ϵ2)+(μ0-ϵ2)ρ2/(m′μ1+ϵ1)ρ-C2(ϵ1,ϵ2)<2γ1.
From now to the end of the present proof, we fix such constants ϵ1, ϵ2, ϵ3, and ρ and denote by |∥uN(τ)∥|G2 the left-hand side of (62). It follows from (62) and (67) that
(68)|∥uN(τ)∥|G2⩽2γ1∫0τ‍|∥u(t)∥|G2dt+1ϵ3∫0τ‍|f(t)|0,G2dt∀τ⩾0.
By the Gronwall-Bellman inequality (44), we deduce from (68) that
(69)|∥uN(τ)∥|G2⩽1ϵ3e2γ1τ∫0τ‍|f(t)|0,G2dt⩽1ϵ3e2(γ1+σ)τ∥f∥L2(Q,σ)2,
for all τ⩾0. Here we used the fact that, for σ⩾0,
(70)∫0τ‍|f(·,t)|0,G2dt=e2στ∫0τ‍|e-στf(·,t)|0,G2dt⩽e2στ∥f∥L2(Q,σ)2.
In the case of σ<0, instead of (61), we give the following inequality
(71)|2Re(f,utN)Qτ|=|2Re(e-σtf,eσtutN)Qτ|⩽ϵ3|eσtutN|0,Qτ2+1ϵ3|e-σtf|0,Qτ2⩽ϵ3|utN|0,Qτ2+1ϵ3|e-σtf|0,Qτ2∀τ⩾0.
Thus, by repeating the above arguments we receive (68) with the last term replaced by the last term of (71), and, therefore, we also get (69) for every real number σ.
Now multiplying both sides of this inequality bye-2(γ+σ)τ, then integrating them with respect to τ from 0 to ∞, we arrive at
(72)|∥uN∥|Q,γ+σ2⩽1ϵ3∥f∥L2(Q,σ)2,
where we used the notation
(73)|∥uN∥|Q,γ+σ2=∫0∞‍e-2(γ+σ)τ|∥uN(τ)∥|G2dτ.
It is clear that |∥·∥|Q,γ+σ is a norm in HBm,1(Q,γ+σ) which is equivalent to the norm ∥·∥HBm,1(Q,γ+σ). Thus, it follows from (72) that
(74)∥uN∥HBm,1(Q,γ+σ)2⩽C∥f∥L2(Q,σ)2,
where C is a constant independent of f and N.
From the inequality (74), by standard weak convergence arguments (see, e.g., [16, Ch. 7]), we can conclude that the sequence {uN}N=1∞ possesses a subsequence convergent to a function u∈HBm,1(Q,γ+σ) which is a generalized solution of problem (8)-(9). Moreover, it follows from (74) that the inequality (22) holds.Now we are going to prove Theorem3. First, we give some needed auxiliary lemmas. The first lemma deals with the regularity of the solution with respect to time variable. It is proved by repeating almost word for word the proof of Theorem 3.4 of [7] with noting that, as in Theorem 2, the assumption σ⩾0 in [7] can be removed.Lemma 7.
Suppose that all the assumptions of Theorem3 are fulfilled. Then utk∈HBm,1(Q,(k+1)γ+σ) for k=0,1,…,h and
(75)∑k=0h‍∥utk∥HBm,1(Q,(k+1)γ+σ)2≤C∑k=0h‍∥ftk∥L2(Q,kγ0+σ)2,
where C is a constant independent of u and f.From the proof of [12, Lemma 5.3] we have the following lemma.Lemma 8.
Letf∈Hm0(G), gj∈Hm2m-μj-1/2(Γ), j=1,…,m, and t0∈ℝ+*. Suppose that u∈HBm(G)∩Hloc2m(G¯∖{0}) is a solution of the following problem:
(76)L(x,t0,∂)u=finG,Bj(x,t0,∂)u=gjonΓ,j=1,…,m.
Then u∈Hm2m(G) and
(77)∥u∥Hm2m(G)2⩽C(∥f∥Hm0(G)2+∑j=1m‍∥gj∥Hm2m-μj-1/2(Γ)2+∥u∥HBm(G)2),
where the constant C is independent of u,f, and t0.Proof of Theorem3.
According to Lemma7 we have
(78)utk∈HBm,1(Q,(k+1)γ+σ),k⩽h.
Moreover, as in proof of Theorem 4.1 of [7], we have
(79)utk(·,t)∈Hloc2m(G¯∖{0})fora.e.t∈ℝ+*andallk⩽h-1.
Now we prove the theorem by induction onh. By (78), utt∈L2(Q,2γ+σ). Thus, from (21) we have
(80)B(t,u,η)=(f(t)-utt(t),η),
for all η∈HBm(G) and a.e. t∈ℝ+*. Since f1(t):=f(t)-utt(t)∈L2(G)⊂Hm0(G) for a.e. t∈ℝ+*, by Lemma 8, we get from (80) that u(t)∈Hm2m(G) for a.e. t∈ℝ+* and
(81)∥u(t)∥Hm2m(G)2⩽C(∥f1(t)∥L2(G)2+∥u(t)∥Hm(G)2),
where C is a constant independent of u,f1, and t. Now multiplying both sides of (81) with e-2(2γ+σ)t, then integrating with respect to t from 0 to ∞ and using estimates from Lemma 7, we obtain
(82)∥u∥Hm2m,1(Q,2γ+σ)2⩽C∥f∥L2(Q,σ)2,
where C is a constant independent of u and f. Hence, the theorem is valid for h=1.
Assume that the theorem is true for some nonnegativeh-2. We will prove it for h-1. Differentiating h-1 times both sides of (80) with respect to t we have
(83)B(t,uth-1,η)=(fth-1(t),η)-(uth+1(t),η)-∑k=0h-2‍(hk)Bth-k-1(t,utk,η),
for all η∈HBm(G), a.e. t∈ℝ+*. From (6) it follows that
(84)Btk(t,v,η)=(Ltk(x,t,∂)v,η),
for all v∈HBm(G)∩Hm2m(G) and η∈HBm(G). Thus, from (83) we deduce
(85)B(t,uth,η)=(Fh(·,t),η),
for all η∈HBm(G) and a.e. t∈ℝ+, where
(86)Fh=fth(·,t)-uth+2(·,t)-∑k=0h-1‍(hk)Lth-k(·,t,D)utk∈Hm0(G).
By the induction assumption, it holds that
(87)utk∈Hm2m,1(Q,(k+2)γ+σ)⊂Hm2m,1(Q,(h+1)γ+σ),k=0,…,h-2.
Moreover,
(88)fth-1∈L2(Q,(h-1)γ+σ)⊂L2(Q,(h+1)γ+σ)
by the assumption of the theorem, and
(89)uth+1∈L2(Q,(h+1)γ+σ)
by Lemma 7. Thus, for a.e. t∈ℝ+*, we have f~(t)∈Hm0(G), g~(t)∈Hαj-(1/2)(Γ), and
(90)∥f~(t)∥Hm0(G)2⩽C(∥fth-1(t)∥2+∥uth+2(t)∥L2(G)2+∑k=0h-2‍∥utk(t)∥Hm2m(G)2),
where C is the constant independent of u, f and t. Now we can repeat the arguments above to conclude that uth-1∈Hm2m,1(Q,(h+1)γ+σ) with the estimate (23) for k=h-1. The proof is completed.


## 5. The Proof of Theorem4

Let us first give some auxiliary lemmas.Lemma 9.
Letl be a nonnegative integer, l⩾2m, and let β1, β2 be real numbers, β1>β2. Suppose that the assumption (H) is fulfilled for the numbers l1=l2=l,β1,β2. Let h∈ℕ, let γ0⩽γ1⩽⋯⩽γh be real numbers, and let f and gj be functions satisfying ftp∈V2,β2l-2m,0(Q,γp), (gj)tp∈V2,β2l-μj-(1/2),0(S,γp) for p=0,…,h. Suppose that u∈V2,β1l,0(Q,γ0) is a solution of the problem
(91)ℒ(t,∂x)u=finQ,ℬj(t,∂x)u=gjonS,j=1,…,m.
Then u admits the following representation:
(92)u(x,t)=∑μ=1N‍∑k=1Λμ‍∑s=0κμ,k-1‍cμ,k,s(t)rλμ(t)×∑σ=0s‍1σ!(lnr)σφk,s(μ)(ω,t)+w(x,t),
where w and cμ,k,s are functions satisfying wtp∈V2,β2l,0(Q,γp), (cμ,k,s)tp∈L2(ℝ+*,γp) for p=0,…,h.Proof.
From the proof of Lemma 4.5 of [13] (see also [17, Lemma 4.1]), it is known that, for each fixed t∈ℝ+*, the solution u(x,t) admits the representation (92) and the following inequality
(93)max(∥wtp(·,t)∥V2,β2l(G)2,|(cμ,k,s)tq(t)|)⩽C({∑j=1m‍∑q=0p‍∥(gj)tq(·,t)∥V2,β2l-μj-12(Γ)2}∑q=0p‍∥ftq(·,t)∥V2,β2l-2m(G)2+∑j=1m‍∑q=0p‍∥(gj)tq(·,t)∥V2,β2l-μj-1/2(Γ)2),
holds for all t∈ℝ+*, p=0,1,…,h, where C is a constant independent of u, f, gj, and t. Now multiplying both sides of (93) by e-2γpt, then integrating with respect to t from 0 to ∞, we see that wtp∈V2,β2l2,0(Q,γp), (cμ,k,s)tp∈L2(ℝ+*,γp). The lemma is proved.By applying Lemma9 and repeating the arguments in the proof of [13, Lemma4.6], we get the following lemma.Lemma 10.
Suppose that all assumptions of Lemma9 are fulfilled. Suppose further that the operators L(x,t,∂x) and Bj(x,t,∂x) are δ-admissible near the origin 0, where δ=β1-β2. Let u∈V2,β1l1,0(Q,γ0) be a solution of the problem
(94)L(x,t,∂x)u=finQ,Bj(x,t,∂x)u=gjonS,j=1,…,m.
Then u admits a representation of the form (92).The following lemma follows directly from the proof of [13, Lemma 4.7].Lemma 11.
Leth∈ℕ, and let γ0⩽γ1⩽⋯⩽γh be real numbers. Let
(95)f=rλ0(t)-2mf~,gj=rλ0(t)-μjg~j,j=1,…,m,
where f~ and g~j are given functions defined on Ω×ℝ+*, ∂Ω×ℝ+, respectively, satisfying (f~)tp∈L2(Ω×ℝ+*,γp),(g~j)tp∈L2(∂Ω×ℝ+,γp), j=1,…,m, p=0,…,h. Suppose that if λ0(t) is an eigenvalue of U(λ,t) for some t, then it is an eigenvalue of U(λ,t) for all t∈ℝ+ with the geometric multiplicity and the partial multiplicities not depending on t∈ℝ+. Then there exists a solution u of the problem (91) which has the form
(96)u=rλ0(t)∑σ=0κ‍1σ!(lnr)σuκ-σ,
where uσ are functions defined on Ω×ℝ+* satisfying (uσ)tp∈L2(ℝ+*,H2m(Ω),γp),σ=0,…,κ. Here κ=0 if λ0(t) is not an eigenvalue of U(λ,t); otherwise κ is the maximal partial multiplicity of λ0(t).Proof of Theorem4.
According to Theorem3, we have
(97)utp∈Hm2m,1(Q,(k+2)γ+σ)⊂Hm+ϵ2m,1(Q,(k+2)γ+σ),
for p=0,…,h-1.
Rewrite (8) in the form
(98)Lu=f1:=f-uttinQ,(99)Bju=0onS,j=1,…,m.
Since utp+2∈L2(Q,(p+2)γ+σ) by (75) and ftp∈L2(Q,pγ+σ) by the assumption, we have
(100)(f1)tp∈L2(Q,(p+2)γ+σ)≡V2,00,0(Q,(p+2)γ+σ)⊂V2,β0,0(Q,(p+2)γ+σ),
for p=0,…,h-1.
Now the assertions of the theorem are obtained by applying Lemma10 and repeating almost word for word the proof of [13, Theorem 4.8].



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