The main results are about the uniqueness and existence of the solutions, and asymptotic expansions of the solution near a neighborhood of conical point

Electronic Journal of Differential Equations, Vol. 2009(2009), No. 151, pp. 1–18.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

Lp-REGULARITY OF SOLUTIONS TO FIRST

INITIAL-BOUNDARY VALUE PROBLEM FOR HYPERBOLIC EQUATIONS IN CUSP DOMAINS

NGUYEN MANH HUNG, VU TRONG LUONG

Abstract. In this article, we establish well-posedness andLp-regularity of solutions to the first initial-boundary value problem for general higher order hyperbolic equations in cylinders whose base is a cusp domain.

1. Introduction

Initial boundary-value problems for hyperbolic and parabolic type equations in a cylinder with base containing conical point have been studied by many authors [8, 9, 10, 13, 14]. The main results are about the uniqueness and existence of the solutions, and asymptotic expansions of the solution near a neighborhood of conical point. Those results are mainly based on Galerkin’s approximate method andL₂-theory.

Boundary-value problems for elliptic type equations and systems have also well studied. The main results, presented in [6, 15, 19, 20], established estimates inLp

for solutions of elliptic boundary value problems in domains with singular points on the boundary.

The question is whether similar results can be obtained based on these results for initial boundary-value problems for non-stationary equations. In this paper, we find the answer for this question.

Firstly, we show the existence of a sequence of smooth domains {Ω}>0 such that Ω⊂Ω and lim→0Ω= Ω. Furthermore, we proved existence, uniqueness and smoothness, with respect to time variable, of the generalized solution by approx- imating boundary method, which can be applied for non-linear equations. Next, by modifying the arguments in [19], we take the term containing the derivative in time of the unknown function to the right-hand side of the equation, such that the problem can be considered as an elliptic problem. With the help of some auxiliary results, we apply the estimates in L_p for solution of the elliptic boundary value problem and our previous estimates to deal with theLp-regularity with respect to both of time and spatial variables of the solution. Finally, in order to illustrate the

2000Mathematics Subject Classification. 35D05, 35D10, 35L30, 35L35, 35G15.

Key words and phrases. Hyperbolic equations; first initial-boundary value problem;

Lp-regularity; cusp domain.

c

2009 Texas State University - San Marcos.

Submitted July 24, 2009. Published November 25, 2009.

1

results above we show an example for the Cauchy-Dirichlet problem for the beam equation in cylinder with base containing a cuspidal point.

2. Preliminaries

Let Ω be bounded domain in Rⁿ, n ≥ 2, with boundary∂Ω. Let p, q be real numbers with 1< p, q <+∞and _p¹+¹_q = 1.

We denote by W_p^m(Ω) the space of all u= u(x), x ∈ Ω that have generalized derivativesD^αu∈Lp(Ω),|α| ≤m. The norm in this space is defined as

u

_m;p=Z

Ω m

X

|α|=0

|D^α_xu|^pdx^1/p .

In particular, ˚W_p⁰(Ω)≡Lp(Ω). The space ˚W_p^m(Ω) is the completion ofC₀^∞(Ω) in norm of the spaceW_p^m(Ω).

Setting Q_T = Ω×(0, T), 0 < T <+∞. We introduce the partial differential operator of order 2m,

L=L(x, t;D_x) =

m

X

|α|,|β|=0

D^α_x

a_αβ(x, t)D_x^β

, (2.1)

whereD^α_x =i^α∂_x^α,aαβ ares×s−matrices of functions with complex values, and aαβ are infinitely differentiable in Q_T and aαβ = a^∗αβ, where a^∗αβ denotes the transposed conjugate matrix ofaαβ. We have the following Green’s formula

Z

Ω

Lu v dx=B[u, v;t]

which is valid for allu, v∈C₀^∞(Ω) and a.e. t∈[0, T), where B[u, v;t] =

m

X

|α|,|β|=0

Z

Ω

aαβ(., t)D_x^βu D^α_xv dx.

We also assume the Garding’s inequality,

B[u, u;t]≥γ₀kuk²_Wm

2 (Ω) (2.2)

which is valid for allu∈W˚₂^m(Ω) and a.e. t∈[0, T), whereγ₀is a positive constant independent ofuandt.

Now we introduce spaces onQT. LetW_p^m,1(QT) be the space consisting of func- tionsu=u(x, t),(x, t)∈QT having generalized derivativesD^αu∈ Lp(QT),|α| ≤ m, andu_t∈L_p(Q_T), with norm

u

_m,1;p=Z

QT

m

X

|α|=0

|D^α_xu|^pdx dt+ Z

QT

|ut|^pdx dt^1/p .

The space ˚W_p^m,1(QT) is the closure inW_p^m,1(QT) of the set consisting of all functions inC^∞(QT), which vanish nearST denoting byC₀^∞(QT) for convenience.

We introduce the spaceW_p^−m,−1(Q_T) of generalized functions onQ_T; it means that iff ∈W_p^−m,−1(QT), thef admits the representation

f = X

|α|≤m

D^α_xf^(α)+f_t^(t) (2.3)

where f^(α), f^(t) ∈ Lp(QT), p ∈ (1,+∞). The norm in W_p^−m,−1(QT) can also be defined by

kfk−m,−1;p= inf X

|α|≤m

kf^(α)kL_p(Q_T)+kf_t^(t)kL_p(Q_T).

Here the infimum is taken over the set of all representations (2.3). It is known that W_p^−m,−1(QT) and ˚W_q^m,1(QT), q=_p−1^p , are dual to one another. We also define

hf, ηi= Z

QT

f η dx dt, f ∈W_p^−m,−1(QT), η∈W˚_q^m,1(QT).

It is clear that

kfk_−m,−1;p= sup{|hf, ηi|:η∈W˚_q^m,1(QT),kηkm,1;q= 1};

kηkm,1;q = sup{|hf, ηi|:f ∈W_p^−m,−1(QT),kfk_−m,−1;p= 1}.

In this paper, we consider the problem

Lu−u_tt=f in Q_T, (2.4)

u= 0, ut= 0 on Ω, (2.5)

∂_ν^ju= 0 onST, j= 0,1, . . . , m−1, (2.6) where f :QT →Cis a given function and∂_ν^juare derivatives with respect to the outer unit normal ofST =∂Ω×(0, T). Setting

B₁[u, η] =

m

X

|α|,|β|=0

Z

Q_T

a_αβD^βu D^αη dx dt+ Z

Q_T

u_tη_tdx dt.

for allu∈W˚_p^m,1(QT), η∈W˚_q^m,1(QT).

Definition 2.1. Let f ∈ W_p^−m,−1(Q_T); a function u is called a generalized L_p- solution of problem (2.4)–(2.6) if and only if u belongs to ˚W_p^m,1(QT), u(x,0) = ut(x,0) = 0, and the equality

B₁[u, η] =hf, ηi (2.7)

holds for allη∈W˚_q^m,1(QT).

To prove uniqueness of the generalized L_p-solution of (2.4) -(2.6), we need to prove the following lemma.

Lemma 2.2. If1< p≤2, then there exists a constantγ2=γ2(p, n, m,|Ω|, T)>0, such that

sup{

B₁[u, η]

:η∈W˚_q^m,1(Q_T),kηk_m,1;q ≤1} ≥γ₂kuk_m,1;p, (2.8) for allu∈W˚_p^m,1(QT).

Proof. We prove this result withu ∈C₀^∞(Q_T). Suppose that there is no γ₂ > 0 such that (2.8) holds. Then there is a sequence{u_k} ⊂C₀^∞(Q_T) withku_kk_m,1;p= 1 and

sup{

B1[uk, η]

:η∈W˚_q^m,1(QT),kηkm,1;q ≤1} ≤ 1

k, for every k≥1. (2.9) Using Garding’ inequality (2.2), we obtain

B1[uk, uk]

≥γ0kukk²_m,0;2+ Z

Q_T

|ukt|²dx dt≥c1kuk²_m,1;2. (2.10)

On the other hand, by using H¨older’s inequality with 1< p <2,p^∗= ²_p,q^∗= _2−p² , we have

ku_kk^p_m,1;p=

m

X

|α|=0

Z

Q_T

|D_x^αu|^pdx dt+ Z

Q_T

|u_t|^pdx dt≤C₂ku_kk^p_m,1;2, (2.11) whereC2=C2(p,|Ω|, T)>0. Combining (2.10) and (2.11), we obtain

B₁[u_k, u_k]

≥Ckukk²_m,1;p,

where c is a constant independent of k. From the above inequality and (2.9), we have

kukk²_m,1;p≤ 1

k C, fork= 1,2, . . . .

which contradicts kukkm,1;p = 1. Therefore, there is a constantγ2 >0 such that (2.8) holds. Since u ∈C₀^∞(QT) which is dense in ˚W_p^m,1(QT), this completes the

proof.

Lemma 2.2 implies the uniqueness of generalized Lp-solution, according to the following theorem.

Theorem 2.3. Assume that coefficients of operator (2.1) satisfy (2.2) and f ∈ W_p^−m,−1(Q_T). Then (2.4)-(2.6)has at most one generalizedL_p-solution.

Proof. Firstly, we prove the theorem in the case 1< p≤2. Suppose that (2.4)-(2.6) has two generalizedLp-solutions u1, u2. Putu=u1−u2, then (2.7) implies that

B1[u, η] =

m

X

|α|,|β|=0

Z

Q_T

aαβ(x, t)D^β_xuD^α_xη dx dt+ Z

Q_T

utη_tdx dt= 0 holds for all η ∈W˚_q^m,1(Q_T). Combining inequality (2.8) with the above equality, we obtain

γ₂kukm,1;p≤sup{

B₁[u, η]

:η∈W˚_q^m,1(Q_T),kηkm,1;q ≤1}= 0.

Next, we prove the theorem in the case p >2. Sincep >2, andQT is bounded, we have ˚W_p^m,1(Q_T) ,→ W˚₂^m,1(Q_T). Therefore, if u is a generalized L_p-solution, and thenuis a generalizedL₂-solution. We obtain the uniqueness of a generalized L_p-solution from the uniqueness of a generalizedL₂-solution. Hence,u≡0 inQ_T.

This completes the proof of theorem.

Next, we prove the approximate boundary lemma, which is the essential tool in solving (2.4) -(2.6).

Lemma 2.4([12]). LetΩbe a bounded domain inRⁿ; then there exists a sequence of smooth domains{Ω^ε}such that Ω^ε⊂Ωandlim_ε→0Ω^ε= Ω.

Proof. Forε > 0 arbitrary, set S^ε ={x∈Ω : dist(x, ∂Ω) ≤ε},Ω^ε= Ω\S^ε and

∂Ω^εis the boundary of Ω^ε. Denote byJ(x) the characteristic function of Ω^ε and byJh(x) the mollification ofJ(x); i.e.,

Jh(x) = Z

Rⁿ

θh(x−y)J(y)dy,

whereθ_h is a mollifier. Ifh < ε/2, thenJ_h(x) has following properties:

(1) J_h(x) = 0 ifx /∈Ω^ε2; (2) 0≤J_h(x)≤1,∀x∈Ω;

(3) Jh(x) = 1 in Ω^2ε; (4) Jh∈C₀^∞(Rⁿ).

We now fix a constant c ∈(0,1), set Ω^ε_c ={x∈Ω : J_h(x)> c}. It is obvious that Ω^ε2 ⊃Ω^ε_c ⊃Ω^2ε. Therefore, Ω^ε_c ⊂Ω and lim_ε→0Ω^ε_c= Ω.

Assume that K is the critical set of J_h, i.e. K consisting of all point x, such that the gradient ofJh atxvanishes. A numberc∈Rsuch that J_h⁻¹(c) contains at least one x ∈ K is called a critical value. By Sard’s theorem then the set of critical values ofJh is of measure zero (see[2, Theorem 1.30]), it implies that there exists a constant c0 ∈ (0,1) such that c0 is not a critical value of Jh. Denote Ω^ε_c0 = {x ∈ Ω : Jh(x) > c0} and F(x) = Jh(x)−c0. For all x⁰ ∈ ∂Ω^ε_c0, then F(x⁰) =Jh(x⁰)−c0= 0 and vector gradJh(x⁰)6= 0. This implies that there exists a ^∂J_∂x^h

i(x⁰)6= 0, without loss of generality we can suppose that ^∂J_∂x^h

n(x⁰)6= 0. Using the implicit function theorem, we obtain that there exists a neighborhood W of (x⁰₁, . . . , x⁰_n−1) inRⁿ⁻¹a neighborhoodVofx⁰_n inRand an infinitely differentiable functionz:W →Rsuch thatx∈ U_x0 ∩∂Ω^ε_c0, where∂Ω^ε_c ={x∈Ω :Jh(x) =c}, Ux⁰ =W × V, if and only if x= (x1, . . . , xn)∈Ux⁰, xn =z(x1, . . . , xn−1). Hence, Ω^ε_c0 is smooth and limε→0Ω^ε_c0 = Ω. The lemma proved.

Suppose that {Ω} is a smooth domain subsequence and lim_ε→0Ω^ε = Ω. Set Q_T = Ω×(0, T), S_T =∂Ω×(0, T). It is known that the problem

Lu−utt=f in Q_T, u= 0, ut= 0 on Ω,

∂_ν^ju= 0 onS_T, j= 0,1, . . . , m−1, has a unique function u(x, t) ∈ C^∞(Q_T); if f ∈ C^∞(Q_T), f_tk

_t=0 = 0, for k = 0,1, . . .. Moreover,u(., t)∈W˚₂^m(Ω), for allt∈[0, T], (see[5, 18, 17]).

3. Main results

3.1. Existence of generalized Lp-solutions. In this subsection, we prove the existence of generalizedLp-solution. Firstly, we prove the needed following propo- sitions:

Proposition 3.1. Suppose that 1< p≤2 andf ∈C^∞(Q_T), andf_tk

_t=0= 0, for k= 0,1, . . . ; thenu is a generalized Lp-solution of (2.4)-(2.5)inQ_T satisfying

kukm,1;p≤Ckfk−m,−1;p

where the constant C is independent of, u andf. Proof. Fromusatisfying system (2.4) in Q_T; i. e.,

f =Lu−u_tt, in Q_T, we have

hf, ηi= Z

Q_T

Luη dx dt− Z

Q_T

u_ttη dx dt valid for allη∈W˚_q^m,1(Q_T).

By using Green’s formula and integrating by parts with respect tot, we obtain from the equality above that

B₁[u, η] =hf, ηi (3.1)

valid for allη∈W˚_q^m,1(Q_T). This clearly shows thatuis a generalizedLp-solutions of problem (2.4) -(2.5) inQ_T; otherwise, using inequality (2.8), we conclude from (3.1) that

kukm,1;p≤Ckfk−m,−1;p.

Now we prove the existence of the generalizedLp-solution of (2.4)-(2.6) in QT, when the assumptions of Proposition 3.1 are satisfied.

Proposition 3.2. Let the following hypothesis be satisfied:

(i) 1< p≤2,

(ii) f ∈C^∞(Q_T), andf_tk

_t=0= 0, for k= 0,1, . . .

Then(2.4)-(2.6)in cylinderQ_T has a generalizedL_p-solutionu∈

◦

W_p^m,1(Q_T)which satisfies

kukm,1;p≤Ckfk_−m,−1;p (3.2)

whereC is a constant independent ofuandf. Proof. By Proposition 3.1 we have

kukm,1;p≤Ckfk−m,−1;p (3.3)

where the constantC does not depend on . Settingue=u in Q_T, and vanishes outsideQ_T. From the inequality above we obtain

kuekm,1;p≤Ckfk−m,−1;p (3.4) where the constantC does not depend on.

It implies that the set {eu^ε}ε>0 is uniformly bounded in the space ˚W_p^m,1(QT).

So we can take a subsequence, denoted also byue^εfor convenience, which converges weakly to a function u ∈ W˚_p^m,1(Q_T). We will show that u is a generalized L_p- solution of (2.4)-(2.6) in cylinderQT. In fact for all η ∈ W˚_q^m,1(QT), there exists η_δ ∈C₀^∞(Q_T) such that η_δ ≡0 inQ_T \Q^ε_T,and kη_δ−ηk_m,1;q →0 when δ→0.

Sinceeu^εis a generalized solution of (2.4)-(2.6) in the smooth cylinderQ^ε_T, we have B₁[ue^ε, η_δ] =hf, η_δi

Passing to the limit whenε→0, δ→0 for the weakly convergent sequence, we get B₁[u, η] =hf, ηi

Since ˚W_p^m,1(QT) is imbedded continuously intoLp(Ω), the trace sequence{ue^ε(x,0)}

of {eu^ε(x, t)} converges weakly to the trace u(x,0) of u(x, t) in Lp(Ω). On the other hand, ue^ε(x,0) = 0, so that u(x,0) = 0;by analogous arguments, we have ut(x,0) = 0. Hence, u(x, t) is a generalized Lp-solution of (2.4)-(2.6). Moreover, from (3.4) we have

kukm,1;p≤lim

ε→0

keu^εkm,1;p≤Ckfk_−m,−1;p.

Proposition 3.2 stated the existence of generalizedLp-solutions of (2.4)-(2.6) in W˚_p^m,1(QT) whenf ∈C^∞(Q_T) andf_tk

_t=0= 0, for k= 0,1, . . .. We now establish the problem whenf ∈W_p^−m,−1(Q_T).

Theorem 3.3. Suppose that f ∈W_p^−m,−1(QT), p∈(1,+∞), then (2.4)-(2.6) has a generalized Lp-solutionu∈W˚_p^m,1(QT), and

kuk^p_m,1;p≤Ckfk^p_0,p, (3.5) whereC is a constant independent ofuandf.

Proof. We start by studying the case 1< p≤2. Denote f_h(x, t) =







0, outsideQ_T f(x, t), t > h

0, t≤h

for allh >0. We denote bygh

2 the mollification offh. Then gh

2 ∈C₀^∞(QT), gh 2 ≡ 0, t < ^h₂ andgh

2 →f inW_p^−m,−1(QT). By Proposition 3.2, problem (2.4)-(2.6) has a generalizedLp-solutionuh∈W^◦

m,1

p (QT) with replacingf bygh

2, and the following estimates holds

ku_hk_m,1;p≤Ckgh

2k_−m,−1;p (3.6)

where C is a constant independent of h, u and f. Since {gh

2} is a Cauchy se- quence in Lp(QT) and inequality (3.6), it follows that {uh} is a Cauchy sequence inW^◦

m,1

p (Q_T). Hence,u_h→u∈W^◦

m,1

p (Q_T), thenuis a generalizedL_p-solutions of (2.4)-(2.6) and satisfies

kukm,1;p≤Ckfk_−m,−1;p. Thus, the theorem is proved in the case 1< p≤2.

Now we study the case p > 2. It is clear that q = _p−1^p ∈ (1,2); by the proof above, for any g ∈ W_q^−m,−1(QT) there exists a solution v ∈ W˚_q^m,1(QT) of the adjoint problem

B1[v, u] =hg, ui (3.7)

for allu∈W˚_p^m,1(QT), and

kvkm,1;q ≤Ckgk_−m,−1;q.

We suppose thatf ∈C^∞(QT),f_tk(x,0) = 0,k= 0,1, . . . and foru=u in (3.7).

Then, by (3.7), we have

|hg, ui|=|B1[v, u]|=|B1[u, v]|=|hf, vi|

≤ kfk_−m,−1;pkvkm,1;q

≤Ckfk_−m,−1;pkgk_−m,−1;q for anyg∈W_q^−m,−1(Q_T). This implies

kukm,1;p= sup |hg, ui|

kgk−m,−1;q

: 06=g∈W_q^−m,−1(QT) ≤Ckfk_−m,−1;p. From this inequality and arguments analogous to proofs above, we get the proof of

the theorem in this case. The proof is complete.

We should remark that by replacing the conditionf ∈W_p^−m,−1(QT) by condition f ∈Lp(QT), and noting that

kfk_W−m,−1

p (Q_T)≤ kfk_Lp(QT), we obtain the following theorem.

Theorem 3.4. If f ∈Lp(QT), p∈(1,+∞), then (2.4)-(2.6), in the cylinder QT, has a generalized Lp-solutionu∈

◦

W_p^m,1(QT) which satisfies kukm,1;p≤Ckfk_Lp(QT) whereC is a constant independent ofuandf.

3.2. Smoothness of the generalizedLp-solution with respect to time. The following theorem shows that the generalized Lp-solutionu∈W˚_p^m,1(QT) of prob- lem (2.4)-(2.6) is smooth with respect to time variablet if right hand-sidef and coefficients of operator (2.1) are smooth enough with respect tot.

Theorem 3.5. Let hbe a positive integer, and assume that (1) f_tk∈L_p(Q_T),k≤h,

(2) f_tk

_t=0= 0, x∈Ω,k≤h−1, (3) sup

∂^ka_αβ

∂t^k

, k < h+ 1 : (x, t)∈QT,0≤ |α|,|β| ≤m ≤µ.

Then the generalized solution u∈W˚_p^m,1(QT)of (2.4)-(2.6)has generalized deriva- tives with respect tot up to orderhinW˚_p^m,1(Q_T)and satisfies the estimate

ku_thk_m,1;p≤c

h

X

k=0

kf_tkk_Lp(QT), (3.8) wherec is a constant independent ofuandf.

Proof. In the case 1< p≤2, Clearly, we needed only to show that ku_thkm,1;p≤

h

X

k=0

kf_tkk_Lp(QT) (3.9) wheref ∈C^∞(QT), f_tk(x,0) = 0, x∈Ω. It is proved by induction onh. According to Proposition 3.1, inequality (3.9) is valid forh= 0. Now let it be true forh−1;

we will prove that this also holds forh.

From the fact thatu satisfies (2.4) inQ_T, we have

f =Lu−u_tt. (3.10)

Differentiating equality (3.10),htimes with respect to t, it follows that f_th =Lu_th+

h−1

X

k=0

h−1 k

D^α_x(a_αβth−kD^β_xu_tk)−u_th+2. Therefore,

hf_th, vi= Z

QT

Lu_thv dx dt+

h−1

X

k=0

h−1 k

Z

QT

m

X

|α|,|β|=0

D_x^α(a_αβth−kD^β_xu_tk)v dx dt

− Z

QT

u_th+1v dx dt

for allv∈W˚_q^m,1(QT).

By using Green’s formula and integrating by parts, B1[u_th, v] =hf_th, vi −

h−1

X

k=0

h−1 k

Z

Q_T m

X

|α|,|β|=0

a_αβth−kD^β_xu_tkD^α_xv dx dt.

for allv∈W˚_q^m,1(QT).

From the inequality above and H¨older’s inequality, we have

|B1[u_th, v]| ≤C(kf_thk_Lp(QT)+

h−1

X

k=0

ku_tkkm,1;p)kvkm,1;q (3.11) for all v ∈ W˚_q^m,1(QT). By using (2.8), (3.11) and the induction assumption, we obtain

ku_thkm,1;p≤C

h

X

k=0

kf_tkk_Lp(QT)

whereC is a constant independent of, u. The proof is completed in this case.

In the case p > 2. It is easy to recognize that q = _p−1^p ∈ (1,2); by Theorem 3.3, for anyg∈W_q^−m,−1(Q_T) there exists a solutionv∈W˚_q^m,1(Q_T) of the adjoint problem

B₁[v, u] =hg, ui (3.12)

which for allu∈W˚_p^m,1(Q_T), and

kvk_m,1;q ≤Ckgk_−m,−1;q.

We assume thatf ∈C^∞(QT),f_tk(x,0) = 0,k= 0,1, . . . and foru=u_th in (3.12).

Then, by (3.12) and (3.11), we have

|hg, u_thi|=|B1[v, u_th]|=|B1[u_th, v]|

≤C(kf_thk_Lp(QT)+

h−1

X

k=0

ku_tkk_m,1;p)kvk_m,1;q

≤C(kf_thk_Lp(QT)+

h−1

X

k=0

ku_tkkm,1;p)kgk_−m,−1;q

for anyg∈W_q^−m,−1(QT). Hence,

ku_thkm,1;p= sup |hg, u_thi|

kgk_−m,−1;q : 06=g∈W_q^−m,−1(QT)

≤C(kf_thkL_p(Q_T)+

h−1

X

k=0

ku_tkkm,1;p).

From this inequality and induction assumption, we have the proof of this case, and

complete the proof.

3.3. Regularity of the generalized Lp-solution. In this section, we consider problem (2.4)-(2.6) in cylindersQT = Ω×(0, T), where its base Ω is described as follows:

Letϕbe an infinitely differentiable positive function on the interval (0,1] satis- fying the conditions

(i) lim_τ→0ϕ(τ)^k−1ϕ(τ)^(k)<∞fork= 1,2, . . .; (ii) R1

0 dτ

ϕ(τ) = +∞

These conditions are satisfied, for example, by the function ϕ(τ) = τ^α if α ≥1.

Obviously, conditions (i) and (ii) imply ϕ(0) = 0. Suppose that Ω is a bounded domain inRⁿ(n≥2), ∂Ω\{O}is smooth, and

{x∈Ω : 0< x_n<1}={x∈Rⁿ:x_n<1, x⁰∈ϕ(x_n)ω},

wherex⁰= (x₁, . . . , x_n−1), ωis a smooth domain inRⁿ⁻¹. Then the mapping yj = xj

ϕ(x_n), ifj= 1, . . . , n−1, and yn= Z 1

xn

dτ

ϕ(τ) (3.13)

takes the set {x∈Ω : 0 < xn <1} onto the half-cylinder C+ ={y ∈ Rⁿ :y⁰ ∈ ω, yn>0}=ω×(0,+∞). Moreover, it follows that

det∂y_j

∂xk

j,k=1,...,n

=ϕ(x_n)⁻ⁿ.

It is known that the function ϕ can be extended to an infinitely differentiable positive function on the interval (0,+∞). To consider the problem, we need to introduce some weighted Sobolev spaces. The space W_p,β,γ^l (Ω) can be defined as the closure of the setC₀^∞(Ω\{O}) with respect to the norm

kuk_Wl

p,β,γ(Ω)=Z

Ω

X

|α|≤l

e^pβyn(xn)ϕ(x_n)^{p(γ−l+|α|)}|D^αu|^pdx1/p

.

LetX, Y be Banach spaces, we denote by Lp(0, T;X) the spaces consisting of all measurable functionsu: (0, T)→X with norm

kukL_p(0,T;X)=Z T 0

ku(t)k^p_Xdt^1/p ,

and byW_p^k(0, T;X, Y), k= 1,2, the spaces consisting of functions u∈L_p(0, T;X) such that generalized derivatives u_tk =u^(k) exist and belong to Lp(0, T;Y), (see [4]), with norm

kuk_Wk

p(0,T;X,Y)=

kuk²_Lp(0,T;X)+

k

X

j=1

kut^jk^p_L

p(0,T;Y)

^1/p .

For short notation, we set

V_p^l(Ω) =W_p,0,0^l (Ω), V_p^l,k(Q_T) =W_p^k(0, T;V_p^l(Ω), L_p(Ω)), W_p,β,γ^l,k (QT) =W_p^k(0, T;W_p,β,γ^l (Ω), Lp(Ω)).

Finally, we define the weighted Sobolev space W_p,β,γ^l (QT) as the set of functions defined inQT such that

kuk_Wl

p,β,γ(Q_T)=Z

Q_T

X

|α|+k≤l

e^2βy(xn)ϕ(xn)p(|γ−l+α|+k)|D^αu_tk|^pdx dt^1/p

<+∞.

To simplify notation, we continue to writeV_p^l(Q_T) instead of W_p,0,0^l (Q_T).

Moreover, we assume that the functions

baαβ(y, .) =ϕ(x(y))^{2m−|α|−|β|}aαβ(x(y), .) (3.14) satisfy the condition of stabilization foryn→+∞for a.e. tin (0, T)(see[19, Sec.9]).

Then the coefficients of the operatorsL(y, t;b D_y), which arises from the operators ϕ(x_n)^2mL(x, t;D_x) via the coordinate change x→ y, stabilize for y_n → +∞. If

we replace the coefficients of the differential operator L(y, t;b Dy) by their limits for yn → +∞, we get differential operator (denote also by L(yb ⁰, t;Dy⁰, Dy_n) for convenience) which has coefficients depending only ony⁰ andt.

By the following proposition, we can apply the results of the Dirichlet problem to elliptic equations in domains with cuspidal points on boundary.

Proposition 3.6. Suppose that u = u(x, t) is a generalized solution of problem (2.4)-(2.6)andutt ∈Lp(QT). Then for a.e. t∈(0, T), u(t) =u(., t)is a general- ized solution in W˚_p^m(Ω) of the Dirichlet problem for elliptic equation

L(., t;D_x)u=f₁(., t) (3.15) wheref1=utt+f.

Proof. For any ψ ∈ W˚_q^m(Ω), θ ∈ C₀^∞(0, T) and setting v(x, t) = ψ(x)θ(t), we substitute the functionv(x, t) into (2.7), we conclude that

Z

Q_T

h X^m

|α|,|β|=0

a_αβD_x^βuD^α_xψ−(u_ttψ+f ψ)i

θ(t)dx dt= 0. (3.16) We will denote by

ξ(t) = Z

Ω

h X^m

|α|,|β|=0

aαβD_x^βuD_x^αψ−(uttψ+f ψ)i dx,

thenξ(t)∈L_p(0, T). Noting thatθ∈C₀^∞(0, T), which dense inL_q(0, T) and using Fubini’s theorem, we obtain from (3.16) that

Z T 0

ξ(t)θ(t)dt= 0, for anyθ∈Lq(0, T),(1/p+ 1/q= 1). (3.17) Therefore,

kξkL_p(0,T)= sup Z T

0

ξ(t)θ(t)dt:θ∈L_q(0, T),kθkL_q(0,T)= 1 = 0.

This impliesξ= 0 for a.e. t∈(0, T). Hence, Z

Ω m

X

|α|,|β|=0

a_αβD_x^βuD_x^αψ dx= Z

Ω

(u_tt+f)ψ dx

for allψ∈W˚_q^m(Ω), for a.e. t∈(0, T). It follows thatu(t) is a generalized solution in W˚_p^m(Ω) of the Dirichlet problem for elliptic equation (3.15), for a.e. t∈(0, T).

In this section, we present the main results which is based on our previous subsection and the results for elliptic equations in cusp domains (cf. [19]). For the start of this section, we denote by U(λ, t)(λ ∈ C, t ∈ (0, T)) the operator corresponding to the parameter-depending boundary-value problem

L(yb ⁰, t;Dy⁰, λ)v= 0 inω;

∂_ν^jv= 0 on∂ω, j= 1, . . . , m−1. (3.18) WhereL(yb ⁰, t;Dy⁰, λ) is the Fourier transformation yn→λofL(yb ⁰, t;Dy⁰, Dy_n).

For each t ∈ (0, T), the operator pencil U(λ, t) is Fredholm, and its spectrum consists of a countable numbers of isolated eigenvalues. The similarly, to Theorem 9.1 in [19], we have the following lemma.

Lemma 3.7. Assume that f1 ∈W_p,β,γ^k (Ω), where β, γ are real numbers. Suppose further that no eigenvalues of U(λ, t), t∈(0, T))line in strip

Imλ₋ ≤Imλ≤Imλ+; Imλ₋ < β < Imλ+

whereλ₊, λ₋ are eigenvalues ofU(λ, t), andImλ₋<0<Imλ₊. Then the gener- alized solution uof the Dirichlet problem for the elliptic equation (3.15),u≡0 if x_n>1, belongs to the space W_p,β,γ^2m+k(Ω) and satisfies the inequality

kuk²_W2m+k

p,β,γ (Ω)≤Ckf1k²_Wk

p,β,γ(Ω) (3.19)

where the constant C is independent off₁.

Proof. Settingωτ=ϕ(τ)ωby the Friederichs inequality, we have Z

ω_τ

|u|^pdx⁰≤Cϕ(τ)^pk X

|γ|=k

Z

ω_τ

|D^γ_x0u|^pdx⁰; therefore,

ϕ(xn)^p(|γ|−m) Z

ω_xn

|D^γ_x0u|^pdx⁰≤C X

|α|=m

Z

ω_xn

|D^α_x0u|^pdx⁰ for all|γ| ≤m. Hence,

X

|γ|≤m

Z

Ω

ϕ(xn)^p(|γ|−m)|D_x^γu|^pdx≤C X

|α|≤m

Z

Ω

|D_x^αu|^pdx (3.20) Letv=v(y) be the function that arises fromϕ(xn)^m−npu(x) via the coordinate change x→y. We setϕ(y_n) =ϕ(x_n), from the properties of the mapping (3.13) and from inequality (3.20), it follows that (ϕ)^−m+npv∈W˚_p^m(C+). Since (ϕ)^−m+npv is the solution of an elliptic equation in C+ with coefficients which stabilize for yn →+∞, i.e.

L(ϕ)b ^−m+npv=fb1

where fb₁ = (ϕ)^2mf₁, we obtain (ϕ)^−m+npv ∈ W˚_p^2m+k(C+) (cf. [19, Theorem 8.1, 8.2]). This impliesu∈W_p,0,m+k^2m+k (Ω). Using the fact that

ϕ(xn)^γ−m+ke^−yn(xn)→0

as x_n →0, if 0< < β, we conclude that u∈ W_p,−,γ^2m+k(Ω). In a similar manner, Theorem 8.2 in [19] it follows thatu∈W_p,β,γ^2m+k(Ω). Furthermore, (3.19) is valid.

Lemma 3.8. Suppose that f, ft ∈ Lp(QT), f(x,0) = 0, and the strip Imλ− ≤ Imλ≤Imλ₊ does not contain eigenvalues ofU(λ, t), t∈(0, T)). Then the gener- alized solution uof problem (2.4)-(2.6),u≡0 if x_n>1, belongs to theV_p^2m,2(Q_T) and satisfies the inequality

kuk_V2m,2

p (QT)≤C[kfk_Lp(QT)+kftk_Lp(QT)], (3.21) where the constant C is independent off.

Proof. Using the smoothness of the generalized solution of (2.4)-(2.6) with respect to t in Theorem 3.5 and Proposition 3.6, we can see that for a.e. t ∈ (0, T), u ∈ W˚_p^m(Ω) is the generalized solution of Dirichlet problem for equation (3.15) with

compact support, wheref1=utt+f ∈Lp(Ω) =W_p,0,0⁰ (Ω) =V_p⁰(Ω). From Lemma 3.7, it implies thatu∈V_p^2m(Ω) for a.e. t∈(0, T) and satisfies the inequality

kuk_V2m

p (Ω)≤C1kf1k_Lp(Ω)≤C kfk_Lp(Ω)+kuttk_Lp(Ω) .

By integrating the inequality above with respect to t from 0 to T, and using the estimates for derivatives of u with respect to t again, we obtain u∈ V_p^2m,2(QT),

which satisfies (3.21).

Theorem 3.9. Let the assumptions of Lemma 3.8 be satisfied, andf_tk∈L_p(Q_T), k ≤2m, f_tk(x,0) = 0, fork = 0,1, . . . ,2m−1. Then the generalized solution u of problem (2.4)-(2.6),u≡0 if x_n >1, belongs to the V_p^2m(Q_T)and satisfies the inequality

kuk_V2m

p (Q_T)≤C

2m

X

k=0

kf_tkk_Lp(QT) (3.22) where the constant C is independent off.

Proof. Let us first prove thatu_ts belongs toV_p^2m,0(Q_T) fors= 0, . . . ,2m−1 and satisfy

ku_tsk_V2m,0

p (Q_T)≤C

2m

X

k=0

kf_tkk_Lp(QT). (3.23) The proof is by done induction ons. According to Lemma 3.8, it is valid fors= 0.

Now let this assertion be true for s−1, we will prove that this also holds for s.

Due to Lemma 3.8 thenusatisfies (2.4), by differentiating both sides of (2.4) with respect tot, stimes, we obtain

Lut^s =ft^s+u_ts+2−

s

X

k=1

s k

L_tku_ts−k (3.24) where

L_tk =L_tk(x, t;D_x) =

m

X

α,β=0

D^α_x∂^kaαβ(x, t)

∂t^k D^β_x .

By the supposition of the theorem and the inductive assumption, the right-hand side of (3.24) belongs to L_p(Q_T). By the arguments analogous to the proof of Lemma 3.8, we getu_ts∈V_p^2m,0(Q_T) and

ku_tsk_V2m,0

p (Q_T)≤C

2m

X

k=0

kf_tkk_Lp(QT) (3.25) whereC is a constant independent ofu, f, ands≤m−1.

Using (3.25) and estimates for derivatives ofuwith respect totin Theorem 3.4, we have

kuk_V2m p (QT)≤

2m−1

X

k=0

ku_tkk_V2m,0

p (QT)+ku_t2mk_Lp(QT)≤C

2m

X

k=0

kf_tkk_Lp(QT). Remark. Letβ be a sufficiently small positive number. Suppose that

e^βyn(xn)f ∈L_p(Q_T), Imλ₋< β <Imλ₊

and the strip

Imλ₋≤Imλ≤Imλ+

contains no eigenvalues of U(λ, t), t ∈ (0, T)); then the generalized solution u of (2.4) -(2.6), u ≡ 0 if xn > 1, belongs to the W_p,β,0^2m (QT). In fact that, setting u= e^−βyn(xn)U, we obtain the first initial boundary value problem which differs little from (2.4)-(2.6). Therefore, U ∈V_p^2m(QT), and thenu∈W_p,β,0^2m (QT). Using the remark above and Lemma 3.7, we obtain the following theorem.

Theorem 3.10. Let the assumptions of Lemma 3.7 be satisfied. Furthermore, we assume thatf_tk ∈W_p,β,γ⁰ (QT), k≤2mandf_tk(x,0) = 0, fork= 0,1, . . . ,2m−1.

Then the generalized solution u of (2.4)-(2.6), such that u≡0 if xn >1, belongs to the W_p,β,γ^2m (QT)and satisfies the inequality

kuk_W^2m

p,β,γ(QT)≤C

2m

X

k=0

kf_tkk_W⁰

p,β,γ(QT) (3.26)

where the constant C is independent off.

This theorem is proved by arguments analogous to those proofs of Lemma 3.8 and Theorem 3.5. Next, we will prove the regularity of the generalized solution of problem (2.4)-(2.6).

Theorem 3.11. Let the assumptions of Lemma 3.7 be satisfied. Furthermore, we assume thatf_tk∈W_p,β,γ^h (QT), k≤2m+handf_tk(x,0) = 0, fork= 0,1, . . . ,2m+ h−1, h∈ N. Then the generalized solution u of (2.4)-(2.6), such that u ≡0 if xn>1, belongs toW_p,β,γ^2m+h(QT)and satisfies the inequality

kuk_W2m+h

p,β,γ (QT)≤C

2m

X

k=0

kf_tkk_Wh

p,β,γ(Q_T) (3.27)

where the constant C is independent ofuandf.

Proof. The theorem is proved by induction on h. Thanks to Theorem 3.9, this theorem is obviously valid for h= 0. Assume that the theorem is true for h−1, we will prove that it also holds forh. It is only needed to show that

ut^s ∈W_p,β,γ^2m+h−s,0(QT) fors=h, h−1. . . ,0;

ku_tsk_W2m+h−s

p,β,γ (Q_T)≤C

2m

X

k=0

kf_tkk_Wh

p,β,γ(Q_T). (3.28) Differentiating both sides of (2.4) again with respect tot, htimes, we obtain

Lu_th =f_th+u_th+2−

h

X

k=1

h k

L_tku_t^h−k (3.29) By the supposition of the theorem and the inductive assumption, the right-hand side of (3.29) belongs to W_p,β,γ⁰ (Ω) for a.e. t ∈ (0, T). Using Lemma 3.7, we conclude thatu_th ∈W_p,β,γ^2m,0(Q_T). It implies that (3.28) holds fors=h. Suppose that (3.28) is true fors=h, h−1, . . . , j+ 1 and setv=u_tj, we obtain

Lv=F_j, (3.30)

whereFj=f_tj+vtt−Pj k=1

j k

L_tku_t^j−k. By the inductive assumption with respect tos,vttbelongs toW_p,β,γ^h−j(Ω) for a.e. t∈(0, T). Thus, the right-hand side of (3.30) belongs to W_p,β,γ^h−j(Ω). Applying Lemma 3.7 again for k = h−j, we get that v = u_tj ∈ W_p,β,γ^2m+h−j(Ω) for a.e. t ∈ (0, T). It means that v = u_tj belongs to W_p,β,γ^2m+h−j,0(Q_T). Furthermore, we have

kvk_W2m+h−j,0

p,β,γ (Q_T)≤CkFjk_Wh−j,0

p,β,γ (Q_T)≤C

2m

X

k=0

kf_tkk_Wh

p,β,γ(QT). (3.31) Therefore,

ku_tjk_W2m+h−j

p,β,γ (Q_T)≤ ku_tj+1k_W2m+h−j−1

p,β,γ (Q_T)+ku_tjk_W2m+h−j,0 p,β,γ (Q_T)

≤C

2m

X

k=0

kf_tkk_Wh p,β,γ(Q_T).

It implies that (3.28) holds fors=j. The proof is complete.

Now we prove the global regularity of the solution.

Theorem 3.12. Let the hypotheses of Lemma 3.7 be satisfied. Furthermore, as- sume thatf_tk ∈W_p,β,γ^h (Q_T), k≤2m+hand f_tk(x,0) = 0, fork= 0,1, . . . ,2m+ h−1,h∈N. Then the generalized solutionuof (2.4)-(2.6)belongs toW_p,β,γ^2m+h(QT) and satisfies the inequality

kuk_W2m+h

p,β,γ (Q_T)≤C

2m

X

k=0

kf_tkk_Wh

p,β,γ(Q_T) (3.32)

where the constant C is independent ofuandf.

Proof. We denote byB the unit ball and suppose thatζ ∈C₀^∞(B), andζ ≡1 in the neighborhood of the originO. We have

L(ζu)−(ζu)tt=ζf+L1u

where L1 is a differential operator, whose coefficients have compact support in a neighborhood of the origin. By Theorem 3.10, we obtain

kζuk_W2m+h

p,β,γ (Q_T)≤C

2m

X

k=0

kf_tkk_Wh p,β,γ(QT).

Setting ζ₁u = (1−ζ)u, then ζ₁u ≡ 0 in a neighborhood of the origin and u = ζu+ (1−ζ)u, and using the smoothness of the solution of this problem in domain with smooth boundary, we get

kζ₁uk_W2m+h

p (Q_T)∼ kζ₁uk_W2m+h

p,β,γ (Q_T)≤C

2m

X

k=0

kf_tkk_Wh p,β,γ(Q_T).

The proof is complete .