Regularity of the Solution of the First Initial-Boundary Value Problem for Hyperbolic Equations in Domains with Cuspidal Points on Boundary

Volume 2009, Article ID 135730,14pages doi:10.1155/2009/135730

Research Article

Regularity of the Solution of

the First Initial-Boundary Value Problem for

Hyperbolic Equations in Domains with

Cuspidal Points on Boundary

Nguyen Manh Hung

and Vu Trong Luong

1_{Department of Mathematics, Hanoi National University of Education, Hanoi 4384, Vietnam}

2_{Department of Mathematics, Taybac University, Sonla city, Sonla 22384, Vietnam}

Correspondence should be addressed to Vu Trong Luong,[email protected]

Received 3 July 2009; Accepted 8 December 2009

Recommended by Martin Schechter

The goal of this paper is to establish the regularity of the solution of the first initial-boundary value problem for general higher-order hyperbolic equations in cylinders with the bases containing cuspidal points.

Copyrightq2009 N. M. Hung and V. T. Luong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Initial boundary-value problems for hyperbolic and parabolic type equations in a cylinder with the base containing conical points have been developed sufficiently by us1–4, the main results of which are about the unique existence of the solution and asymptotic expansions of the solution near a neighborhood of a conical point. However, those problems mentioned above in cylinder with base containing cuspidal point, also interesting for applied sciences, have not been studied yet.

In the present paper, we are concerned with the first initial boundary value problems for higher hyperbolic equation in a cylinder, whose base containing cuspidal points.

In5,6we showed the existence of a sequence of smooth domains{Ω_}

>0such that Ω _{⊂ Ω and lim}

→0Ω Ω. Furthermore, we proved the existence, the uniqueness, and

Our paper is organized as follows: inSection 2, we introduce exterior cusp domain and weight Sobolev spaces. InSection 3, we will state the formulation of the problem. The main results, Theorems4.3,4.6, and4.7, are stated in Section 4, and examples are given in

Section 5.

2. Cusp Domain and Weighted Sobolev Spaces

Let ϕ be an infinitely differentiable positive function on the interval 0,1 satisfying the following conditions:

i limτ→0ϕτ k−1ϕτ k <∞fork1,2, . . . ,

ii 1₀dτ/ϕτ ∞.

These conditions are satisfied; the functionϕτ τα _{if α ≥ 1 is an example. Obviously,}

conditionsi andii implyϕ0 0. We assume thatΩis a bounded domain inRn_{, ∂Ω\{O}}

is smooth, and

{x∈Ω:xn<1}

x∈Rn:xn<1, x∈ϕxn ω

, 2.1

wherex x1, . . . , xn−1 andωis a smooth domain inRn−1. Then the mapping

yj

xj

ϕxn , j

1, . . . , n−1,

yn ₁

xn dτ ϕτ

2.2

takes the set{x ∈ Ω : xn < 1}onto the half-cylinder C {y ∈ Rn : y ∈ ω, yn > 0}

ω×0,∞ . Moreover, it follows that

det

∂yj

∂xk

j,k1,...,n

ϕxn −n. 2.3

We extend the functions ϕ to an infinitely differentiable positive function on the interval

0,∞ .The spaceHl

β,γΩ can be defined as the closure of the setC∞Ω\ {O} with respect

to the norm

u_Hl β,γΩ

⎛ ⎝

Ω

|α|≤l

e2βynxn _ϕx

n 2γ−l|α||Dαu|2dx ⎞ ⎠

1/2

. 2.4

It is known thatu∈Hl

We also denote byHl_{Ω the Sobolev space of functionsuux andx∈Ωthat have}

generalized derivativesDα_{u∈L2Ω ,|α| ≤l. The norm in this space is defined as follows:}

u_Hl_Ω ⎛ ⎝

Ω

m

|α|0

|Dα_u|2_dx

⎞ ⎠

1/2

. 2.5

The space

◦

Hl_{Ω is the completion ofC}∞

0 Ω in norm of the spaceHlΩ .

SetQT Ω×0, T ; we proceed to introduce some functional spaces. LetX, Ybe Banach

spaces, we denote byL20, T;X the spaces consisting of all measurable functionsu:0, T → Xwith norm

uL20,T;X

T

0

ut 2Xdt 1/2

, 2.6

and byHk_{0, T;X, Y ,k1,2,the spaces consisting of all functionsu∈L20, T;X such that}

generalized derivativesutk uk exist and belong toL20, T;Y ,see9 , with norms

uHk_0,T;X,Y ⎛ ⎝_u2

L20,T;X

k

j1

utj2_L

20,T;Y

⎞ ⎠

1/2

. 2.7

For shortness, we set

Vl_{Ω H}l

0,0Ω , Vl,kQT Hk

0, T;Vl_{Ω , L2Ω ,}

Hl,1_Q T H1

0, T;Hl_{Ω , L2Ω , H}l,k

β,γQT Hk

0, T;H_β,γl Ω , L2Ω . 2.8

Finally, we define the weighted Sobolev spaceHl

β,γQT as a set of all functions defined inQT

such that

uHl β,γQT

⎛ ⎝

QT

|α|k≤l

e2βyxn _ϕx

n 2|γ−lα|k|Dαutk|2dx dt ⎞ ⎠

1/2

<∞. 2.9

To simplify notation, we continue to writeVl_Q

T instead ofH₀l_,0QT .

3. Formulation of the Problem

Let us consider the partial differential operator of order 2m

Lx, t;Dx m

|α|,|β|0

Dαx

aαβx, t Dβx

where aαβ are functions with complex values,aαβ −1 |α||β|a∗αβ a∗αβ denotes the

trans-posed conjugate ofaαβ andaαβare infinitely differentiable inQT. Moreover, we assume that

the functions

aαβ

y,·ϕxy2m−|α|−|β|aαβ

xy,· 3.2

satisfy the condition of stabilization foryn → ∞for a.e.tin0, T see7, Section 5.5 . Then

the coefficients of the operatorsLy, t; Dy , which arise from operatorsϕxn 2mLx, t;Dx via

the coordinate changex → y, stabilize foryn → ∞. If we replace the coefficients of the

differential operatorLy, t; Dy by their limits foryn → ∞, we get differential operator

which has coefficients depending only onyandtfor the convenience in use, we denote also byLy, t; Dy, D_yn .

In the paper, we usually use the following Green’s formula:

ΩLx, t; Dx u v dxBu, v;t 3.3

which is valid for allu, v∈C∞₀ Ω and a.e.t∈0, T , where

Bu, v;t

m

|α|,|β|0 −1 |α|

Ωaαβ·, t D

β

xu Dαxv dx. 3.4

We also suppose that the form B·,·;t is Hm_{Ω -elliptic uniformly with respect to}

t∈0, T , that is, the inequality

−1 mBu, u;t ≥γ0u2Hm_Ω 3.5

is valid for allu∈H◦m_{Ω and allt∈0, T , whereγ}

0is the positive constant independent of

uandt. In this paper, we consider the following problem:

−1 m−1Lu−uttf inQT, 3.6

u0, ut0 on Ω, 3.7

∂jνu0 onST, j0,1, . . . , m−1, 3.8

where f ∈ L2QT and ∂jνuare derivatives with respect to the outer unit normal of ST

Definition 3.1. A functionuis called a generalized solution of problem3.6 –3.8 if and only ifubelongs toH◦m,1_Q

T , ux,0 0,and the equality

−1 m−1

m

|α|,|β|0 −1 |α|

QT

aαβDβx uDαxη dx dt

QT

utηtdx dt

QT

fη dx dt 3.9

holds for allη∈H◦m,1_Q

T , ηx, T 0.

The existence, the uniqueness and the smoothness with respect to the time variable

for the generalized solution of problem 3.6 –3.8 in the Sobolev space H◦m,1_Q

T were

established in5,6according to following theorems:

Theorem 3.2. Assume thatf∈L2QT ,and there exists a positive numberμsuch that

sup∂aαβ

∂t

,aαβ:x, t ∈QT,0≤ |α|,β≤m

≤μ. 3.10

Then problem 3.6, 3.8 has the unique generalized solution u ∈ Hm,1_Q

T , and the following estimate holds

u2

m,1≤Cf 2

L2QT , 3.11

whereCis a constant independent ofuandf.

Theorem 3.3. Suppose that the following hypotheses are satisfied:

i ∂k_f/∂tk_∈L2Q

T , k≤h;

ii ∂kf/∂tk|t00, x∈Ω, k ≤h−1;

iii sup{|∂k_a

αβ/∂tk|, k < h:x, t ∈QT,0≤ |α|,|β| ≤m} ≤μ.

Then the generalized solutionu∈Hm,1_Q

T of problem3.6,3.8 has generalized derivatives with respect totup to order h inHm,1_Q

T and satisfies the following estimate:

uth2_m,1≤C

h

k0

_ftk2_L

2QT , 3.12

whereCis a constant independent ofuandf.

Proposition 3.4. Suppose thatuux, t is a generalized solution of problem3.6 –3.8 andutt∈

L2QT . Then for a.e.t ∈ 0, T ,ut u·, t is a generalized solution in

◦

Hm_{Ω of the Dirichlet} problem for elliptic equation

L·, t;Dx uf1·, t , 3.13

wheref1 −1 m−1uttf .

Proof. Let {ψk}∞k1 be an orthogonal basis of the space

◦

Hm_{Ω . Setting ηx, t ψ}

kx θt ,

whereθ∈C∞₀ 0, T , and substituting the functionηx, t into3.9, we conclude that

QT ⎡ ⎣ m

|α|,|β|0

−1 |α|aαβDβx uDαxψk −1 m

utt ψkfψk ⎤_⎦

θt dx dt0. 3.14

We will denote by

ξt

Ω ⎡ ⎣ m

|α|,|β|0

−1 |α|aαβDxβuDxαψk −1 m

uttψkfψk ⎤_⎦

dx, 3.15

thatξt ∈ L20, T .Noting thatθ ∈ C₀∞0, T and using Fubini’s theorem, we obtain from

3.14 thatξ 0 in0, T \Ek, whereEk is a set of measure zero. Since{ψk}∞k1 are dense in

◦

Hm_{Ω , the following equality}

Ω

m

|α|,|β|0

−1 |α|aαβDxβuDxαψ dx −1 m−1

Ω

uttf

ψ dx 3.16

holds for allψ ∈H◦m_{Ω ,for allt∈0, T \}∞

k1Ek. It follows thatut is a generalized solution

inH◦m_{Ω of the Dirichlet problem for elliptic equation3.13 , for a.e.t∈0, T .}

4. The Main Results

In this section, we would like to present the main results of the study which is based on our previous resultscf.5,6 and the results of elliptic equations in cusp domainscf.7 . For the start of this section, we denote byUλ, t λ∈C, t∈0, T the operator corresponding to the parameter-depending boundary value problem

Ly, t; Dy, λu0 in ω; ∂j_νu0 on ∂ω, j1, . . . , m−1. 4.1

Lemma 4.1. Assume that f1 ∈ H_β,γk Ω , where β, γ are real numbers. Additionally, the authors

suppose that no eigenvalues ofUλ, t , t ∈0, T line in stripsReλ− ≤ Reλ ≤ Reλ andReλ− <

β <Reλ, whereλandλ−are eigenvalues ofUλ, t, andReλ− <0<Reλ. Then the generalized solutionuof the Dirichlet problem for elliptic equation3.13 , such thatu≡0ifxn >1, belongs to theH_β,γ2mkΩ and satisfies the inequality

u2 H2mk

β,γ Ω ≤C _f12

Hk

β,γΩ , 4.2

where the constantCis independent off1.

Proof. Setting

ωτ ϕτ ω 4.3

by the Friederichs inequality, we have

ωτ

|u|2_dx≤Cϕτ 2k

|γ|k

ωτ Dγ_xu

2

dx; _4.4

therefore,

ϕxn 2|γ|−m

ωxn Dγ_xu

2

dx≤C

|α|m

ωxn _Dα

xu2dx _4.5

for all|γ| ≤m. Hence,

|γ|≤m

Ωϕxn

2|γ|−m

Dγxu 2

dx≤C

|α|≤m

Ω|D

α

xu|2dx. 4.6

Let v vy be the function which arises fromϕxn m−n/2ux via the coordinate

changex → y. We setϕyn ϕxn ; then from the properties of the mapping2.2 and from

inequality4.6, it follows thatϕ −mn/2v ∈ Hm_{C . Sinceϕ} −mn/2_{vis the solution of an}

elliptic equation inCwith coefficients which stabilize foryn → ∞, that is,

Lϕ−mn/2vf1, 4.7

wheref1 ϕ 2mf1, we obtainϕ −mn/2v ∈ H2mkC cf.7, Lemma 5.5.3 . This implies

u∈H₀2_,mmk_kΩ . Using the fact that

ϕxn γ−mke−ynxn −→0, 4.8

asxn → 0,if 0< < β, we conclude thatu∈H−2m,γkΩ .From Corollary 9.1.1 in7it follows

thatu∈H2mk

Lemma 4.2. Suppose thatf, ft∈L2QT , fx,0 0, and the stripReλ− ≤Reλ≤Reλdoes not

contain eigenvalues ofUλ, t , t∈0, T . Then the generalized solutionuof problem3.6 –3.8 , such thatu≡0ifxn>1, belongs to theV2m,2QT and satisfies the inequality

u2 V2m,2_Q

T ≤C

f2_L

2QT ft

2 L2QT

, 4.9

where the constantCis independent off.

Proof. Using the smoothness of the generalized solution of problem3.6–3.8 with respect totinTheorem 3.3andProposition 3.4, we can see that for a.e.t ∈0, T , u∈ H◦m_{Ω is the}

generalized solution of Dirichlet problem for3.13 with f1 uttf ∈L2Ω H00,0Ω

V0Ω . From Lemma 4.1, it implies that u ∈ V2mΩ for a.e. t ∈ 0, T and satisfies the inequality

u2

V2m_Ω ≤C1f12_L2Ω ≤C

f2_L

2Ω utt

2 L2Ω

. 4.10

By integrating the inequality above with respect totfrom 0 toT, and using the estimates for derivatives ofuwith respect totagain, we obtainu∈V2m,2_Q

T , which satisfies inequality 4.9 .

Theorem 4.3. Let the assumptions ofLemma 4.2be satisfied, andftk ∈L2Q_T ,k≤2m,f_tkx,0

0,fork0,1, . . . ,2m−1. Then the generalized solutionuof problem3.6 –3.8 , such thatu≡0if xn>1, belongs to theV2mQT and satisfies the inequality

u2

V2m_QT ≤C

2m

k0

_ftk2_L

2QT , 4.11

where the constantCis independent off.

Proof. Let us first prove thatutsbelong to theV2m,0Q_T fors0, . . . ,2m−1 and satisfy

uts2_V2m,0_Q

T ≤C

2m

k0

_ftk2_L

2QT . 4.12

The proof is an induction ons. According toLemma 4.2, it is valid for s 0. Now let this assertion be true for s−1; we will prove that this also holds for s. Due to Lemma 4.2, u

satisfies3.6. By differentiating both sides of3.6 with respect tot, stimes, we obtain

Luts −1 m−1

ftsu_ts2

−1 m

s

k1

s k

where

Ltk L_tkx, t;D_x

m

α,β0

Dαx

∂k_a αβx, t

∂tk D β x

. 4.14

By the supposition of the theorem and the inductive assumption, the right-hand side of

4.13 belongs to L2QT . By the arguments analogous to the proof ofLemma 4.2, we get

uts ∈V2m,0Q_T and

uts2_V2m,0Q T ≤C

2m

k0

_ftk2_L

2QT , 4.15

whereCis a constant independent ofu, f, ands≤m−1.

By using4.15 and estimates for derivatives ofuwith respect totinTheorem 3.3, we have

u2 V2m_Q

T ≤

2m−1

k0

utk2_V2m,0_QT ut2m2_L

2QT

≤C

2m

k0

_ftk2_L

2QT .

4.16

Remark 4.4. Letβbe a sufficiently small positive number. Suppose that eβynxn _{f ∈ L2QT ,}

and the strip Reλ− ≤ Reλ ≤ Reλ contains no eigenvalues of Uλ, t , t ∈ 0, T ; then

the generalized solutionuof problem3.6 –3.8 , such thatu ≡ 0 ifxn > 1, belongs to the

H_β,2m₀QT . In fact, settingu e−βynxn U, we obtain the first initial boundary value problem

which differs little from3.6–3.8. Therefore,U∈V2mQT , and thenu∈H_β,2m₀QT . Using

the remark above andLemma 4.1, we obtain the following theorem.

Theorem 4.5. Let the assumptions ofLemma 4.1be satisfied. Furthermore, the authors assume that ftk ∈H_β,γ0 Q_T ,k ≤2m,andf_tkx,0 0,fork0,1, . . . ,2m−1. Then the generalized solutionu of problem3.6 –3.8 , such thatu≡0ifxn>1, belongs to theH_β,γ2mQT and satisfies the inequality

u2 H2m

β,γQT ≤C

2m

k0

_ftk2

H0

β,γQT , 4.17

where the constantCis independent off.

This theorem is proved by arguments analogous to those proofs ofLemma 4.2and

Theorem 4.3. Next, we will prove the well regularity of the generalized solution of problem

Theorem 4.6. Let the assumptions ofLemma 4.1be satisfied. Furthermore, the authors assume that ftk ∈ H_β,γh Q_T ,k ≤ 2mh, and f_tkx,0 0,fork 0,1, . . . ,2mh−1, h ∈ N. Then the generalized solutionuof problem3.6–3.8, such thatu≡ 0ifxn >1, belongs to theH_β,γ2mhQT and satisfies the inequality

u2 H2mh

β,γ QT ≤C

2m

k0

_ftk2_Hh

β,γQT , 4.18

where the constantCis independent ofuandf.

Proof. The theorem is proved by induction on h. Thanks to Theorem 4.5, this theorem is obviously valid forh 0. Assume that the theorem is true forh−1, we will prove that it also holds forh. It is only needed to show that

uts∈H2mh−s,0

β,γ QT forsh, h−1, . . . ,0,

uts2

H2mh−s

β,γ QT ≤C

2m

k0

_ftk2

Hh β,γQT .

4.19

Differentiating both sides of3.6 again with respect tot, htimes, we obtain

Luth −1 m−1

fthu_th2

−1 m

h

k1

h k

Ltku_th−k. 4.20

By the supposition of the theorem and the inductive assumption, the right-hand side of4.20

belongs toH0

β,γΩ for a.e.t∈0, T . UsingLemma 4.1, we conclude thatuth ∈H_β,γ2m,0Q_T . It

implies that4.19 holds forsh. Suppose that4.19 is true forsh, h−1, . . . , j1,and set

vutj, we obtain

LvFj, 4.21

where Fj −1 m−1ftj v_tt −1 mj

k1

_j

k

Ltku_tj−k. By the inductive assumption with

respect tos, vtt belongs to H_β,γh−jΩ for a.e. t ∈ 0, T . Thus, the right-hand side of 4.21

belongs toH_β,γh−jΩ . ApplyingLemma 4.1again forkh−j, we get thatvutj ∈H2mh−j

β,γ Ω

for a.e.t∈0, T . It means thatvutjbelongs toH2mh−j,0

β,γ QT . Furthermore, we have

v2

H2_β,γmh−j,0QT ≤CFjH_β,γh−j,0QT ≤C

2m

k0

_ftk2_Hh

Therefore,

utj2

H_β,γ2mh−jQT ≤ utj1 2

H_β,γ2mh−j−1QT utj 2

H2_β,γmh−j,0QT

≤C

2m

k0

_ftk2

Hh β,γQT .

4.23

It implies that4.19 holds forsj. The proof is complete.

Now we will prove the global regularity of the solution.

Theorem 4.7. Let the hypotheses ofLemma 4.1be satisfied. Furthermore, supposeftk ∈ Hh

β,γQT ,

k≤2mh,andftkx,0 0,fork0,1, . . . ,2mh−1, h∈N. Then the generalized solutionuof problem3.6 –3.8 belongs to theH_β,γ2mhQT and satisfies the inequality

u2 H2mh

β,γ QT ≤C

2m

k0

_ftk2_Hh

β,γQT , 4.24

where the constantCis independent ofuandf.

Proof. We denote by B the unit ball, and suppose that ζ ∈ C∞₀ B , and ζ ≡ 1 in the neighborhood of the originO. It is easy to get that

−1 m−1Lζu −ζu ttζfL1u, 4.25

where L1 is a differential operator, whose coefficients have compact support in a

neighborhood of the origin. By arguments analogous to the proof ofTheorem 4.6, we obtain

ζu2 H2mh

β,γ QT ≤C

2m

k0

_ftk2_Hh

β,γQT . 4.26

Setζ1u 1−ζ u, thenζ1u≡0 in a neighborhood of the origin anduζu1−ζ u, and using

the smoothness of the solution of this problem in domain with smooth boundary, we get

ζ1u2_H2mh_Q

T ∼ ζ1u

2 H2mh

β,γ QT ≤C

2m

k0

_ftk2_Hh

β,γQT . 4.27

5. Examples

In this section, we apply the results of the previous section to the Cauchy-Dirichlet problem for the wave equation. The assumptions can be described as follows:Ωis a bounded domain inRn_{, ∂Ω\ {O}is smooth,}

{x∈Ω: 0< xn<1} ≡

x∈Rn: 0< xn<1,x< ϕxn

, 5.1

wherex x1, . . . , xn−1 ,ϕ∈C∞0,1 ,ϕxn → 0,ϕxn ϕxn → 0 asxn → 0 andϕ0 0,

andQT Ω×0, T ,ST ∂Ω\ {O} ×0, T .

We consider the Cauchy-Dirichlet problem for the wave equation inQT:

Δu−uttf inQT,

u0, ut0 onΩ,

u0 onST, j 0,1, . . . , m−1,

5.2

wheref ∈ H0

β,γQT . It follows the results ofSection 4that if|β| < z, wherez is the least

positive root of the Bessel functionJn−3/2z , then problem5.2 has a unique solutionuin

H_β,γ2 QT and we have the estimate

u2 H2

β,γQT ≤C exp

β

1

xn dτ ϕτ

ϕγxn f

2

L2QT

. 5.3

Moreover, ifftk ∈H_β,γh Q_T , k≤2h, andf_tkx,0 0 fork0,1, . . . ,1h, thenu∈H_β,γ2hQ_T

and satisfies

u2 H2h

β,γQT ≤C

2

k0

_ftk2_Hh

β,γQT . 5.4

For the two-dimensional casen2 , and lettingϕτ τ2_{, we consider problem5.2}

in the cylinderQT Ω×0, T , whereΩis a bounded domain inR2, ∂Ω\ {O}is smooth, and

x, y∈Ω: 0< x <1≡x, y∈R2: 0< x <1,y< x2. 5.5

Thus, the change of variables

ξ ₁

x

dτ τ2 x

−1_{−1, ηyx}−2 _5.6

transforms

With notationsvξ, η, t ux, y, t, we have

ux, yvx−1_{−1, yx}−2_{, 5.8}

∂yux−2∂ηv, ∂2yyux−4∂ηη2 v, ∂xu−x−2∂ξv−2yx−3∂ηv,

∂2

xxux−3∂ξv6yx−4∂ηvx−4∂2_ξξv4yx−5∂2_ξηv4y2x−6∂2ηηv

x−4x∂ξv6y∂ηv∂2ξξv4yx−1∂2ξηv4y2x−3∂2ηηv

x−4∂2_ξξv4ηξ1 −1∂2_ξηv4η2ξ1 −2∂2ηηv

ξ1 −1∂ξv6ηξ1 −2∂ηv

.

5.9

Hence, the differential operatorΔ, which arises from the differential operatorx4_{Δuvia the}

coordinate changex, y → ξ, η , turns out to be

Δv∂2

ξξv∂2ηηv4ηξ1 −1∂2ξηv4η2ξ1 − 2_∂2

ηηv

ξ1 −1∂ξv6ηξ1 −2∂ηv.

5.10

Clearly, coefficients of differential operatorΔ stabilize forξ → ∞, and the limit differential operator ofΔ denoted byΔ for convenience is

Δv∂2_ξξv∂2ηηv. 5.11

We denote also byUλ λ ∈ C ,the operator corresponding to the parameter-depending boundary value problem

vηηλ2v0, v−1 v1 0. 5.12

Eigenvalues ofUλ are roots of the Bessel function

J−1/2z

2

πz !1/2

cosz; 5.13

Jν, ν >−1 has only real rootssee10, Theorem 1, page 94 . Therefore, they are

λk π₂ kπ, k∈Z. 5.14

It is easy to see thatλ π/2 is the least positive root of the Bessel functionJ−1/2z . From

Theorem 5.1. Suppose thatftk ∈H_β,γ0 Q_T ,k≤2,|β|< π/2,γis a real number andf_tkx,0 0, fork0,1. Then problem5.2 has a unique solutionuinH2

β,γQT ,and we have the estimate

u2 H2

β,γQT ≤C

2

k0

eβ1/x−1x2γftk

2

L2QT

. 5.15

Moreover, ifftk ∈H_β,γh Q_T , k≤2h, andf_tkx,0 0fork0,1, . . . ,1h, thenu∈H_β,γ2hQ_T and satisfies

u2 H2h

β,γQT ≤C

2

k0

_ftk2_Hh

β,γQT . 5.16

In case that boundary ofΩhas some cuspidal points, then by arguments analogous to

Section 4, we consequently obtain the similar results.

Acknowledgment

This work was supported by National Foundation for Science and Technology Development

NAFOSTED , Vietnam.

References

1 N. M. Hung, “The first initial boundary value problem for Schr ¨odinger systems in non-smooth domains,”Differentsial’nye Uravneniya, vol. 34, pp. 1546–1556, 1998Russian.

2 N. M. Hung, “Asymptotic behavior of solutions of the first boundary value problem for strongly hyperbolic systems near a conical point of the domain boundary,”Matematicheski˘ıSbornik, vol. 190, no. 7, pp. 103–126, 1999.

3 N. M. Hung and N. T. Anh, “Regularity of solutions of initial-boundary value problems for parabolic equations in domains with conical points,”Journal of Differential Equations, vol. 245, no. 7, pp. 1801– 1818, 2008.

4 N. M. Hung and J.-C. Yao, “On the asymptotics of solutions of the first initial boundary value problem to hyperbolic systems in infinite cylinders with base containing conical points,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1620–1635, 2009.

5 N. M. Hung and V. T. Luong, “Unique solvability of initial boundary-value problems for hyperbolic systems in cylinders whose base is a cusp domain,”Electronic Journal of Differential Equations, vol. 2008, no. 138, pp. 1–10, 2008.

6 V. T. Luong, “On the first initial boundary value problem for strongly hyperbolic systems in non-smooth cylinders,”Journal of S. HNUE, vol. 1, no. 1, 2006.

7 V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, vol. 52 ofMathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1997.

8 V. G. Maz’ya and B. A. Plamenevski˘ı, “Estimates inLpand in H ¨older classes, and the Miranda-Agmon

maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary,”Mathematische Nachrichten, vol. 81, pp. 25–82, 1978, English translation in: American Mathematical Society Translations, vol. 123, pp. 1–56, 1984.

9 L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1998.