Abstract and Applied Analysis

Volume 2013 (2013), Article ID 801314, 10 pages

http://dx.doi.org/10.1155/2013/801314

Research Article

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

Nguyen Manh Hung and Nguyen Thanh Anh

Department of Mathematics, Hanoi National University of Education, No. 136 Xuan Thuy Street, Hanoi, Vietnam

Received 24 December 2012; Accepted 14 July 2013

Academic Editor: Ağacik Zafer

Copyright © 2013 Nguyen Manh Hung and Nguyen Thanh Anh. 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.

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.

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. Section 2 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

Let be a bounded domain in with the boundary . We suppose that is a smooth manifold and in a neighborhood of the origin coincides with the cone , where is a smooth domain on the unit sphere in . For each , , denote ,  . Specifically, we set , , where , . For each multiindex , set and . For a nonnegative integer we write instead of .

We introduce the following differential operator:

where are bounded complex-valued functions defined in . We assume that for all . This means the differential operator is formally self-adjoint. We assume further that there exists a positive constant such that

for all , and all .

We introduce also a system of boundary operators:

on with smooth coefficients in . Suppose that

and the coefficients of are independent of if , where stands for the order of the differential operator . Suppose that is a normal system on for all ; that is, the two following conditions are satisfied:

(i) for ,(ii) for all , .

Here is the unit outer normal to at point , and is the principal part of ,

Furthermore, we assume that for all sufficiently close to the origin .

To be able to reduce the problem considered to variational form we assume that there are boundary operators on , , such that

for all and a.e. . Here

is the bilinear form associated with the operator . 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 are less than ) or a generalized Neumann system (then for all ), the equality (6) holds for a suitable system (see [10, Section I.7.]).

In this paper, we consider the following problem:

Before giving the definition of generalized solutions to this problem, let us introduce some needed functional spaces.

Let us denote by the usual Sobolev space of all functions defined in with the norm

If , by we denote the space of traces of functions in on with the norm

We set

with the same norm as in . By we denote the closure of with respect to the norm

where . By we denote the weighted Sobolev space of functions defined in with the norm

If , then denote the spaces consisting of traces of functions from respective spaces on the boundary with the respective norms

Let be Banach spaces, . We denote by the space of all functions such that

and by the space of all functions such that . The norm in is defined by

For some , we denote by the space of all functions such that

and by the space of all functions such that with the norm

For shortness, we set

Now the definition of generalized solutions of the problem (8)-(9) is given as follows.

Definition 1. Let be a given function defined on which belongs to for each . A function for some is called a generalized solution of the problem (8)-(9) if and only if , and for each the equality

holds for all satisfying .

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 number such that for each , if for some real number , the problem (8)-(9) has a unique generalized solution in the space and

where is a constant independent of and .

The following theorem states on the regularity of the generalized solution in weighted Sobolev spaces.

Theorem 3. Let be a positive integer and be the number as in Theorem 2. Suppose that the function satisfies the following conditions for some real number :

(i), (ii).

Let for some be the generalized solution of the problem (8)-(9). Then for and

where is a constant independent of and .

The proofs of these theorems will be given in Section 4. The number 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.

Let and be an arbitrary local coordinate system on . Let be a positive real constant. A differential operator

is called -admissible operator of order near the conical point if the coefficients are infinitely differentiable in and there is representation in a neighborhood of the conical point :

where and the functions are infinitely differentiable functions in such that

for every multiindex and every pair of nonnegative integers . Here the constants do not depend on and . The leading part of the operator at the point is defined by

It can be directly verified that the derivative has the form

where are differential operators of order with smooth coefficients on . Thus the operator can be represented as

For convenience we rewrite the operator in the form

Let be the leading parts of , at the point . Since the coefficients of the operators and are smooth, it is verified easily that

and the operators and are -admissible. Rewrite in the form

We introduce the operator

() of the parameter-depending elliptic boundary value problem

This is a pencil of continuous operators from

depending polynomially on .

We mention now some well-known definitions [3]. Let fixed. If , such that , then is called an eigenvalue of and is called an eigenvector corresponding to . is called the geometric multiplicity of the eigenvalue .

If the elements of satisfy the equations

then the ordered collection is said to be a Jordan chain corresponding to the eigenvalue of the length . The rank of the eigenvector () is the maximal length of the Jordan chains corresponding to the eigenvector .

A canonical system of eigenvectors of corresponding to the eigenvalue is a system of eigenvectors such that is maximal among the of all eigenvectors corresponding to and is maximal among the of all eigenvectors in any direct complement in to the linear span of the vectors . The numbers are called the partial multiplicities and the sum is called the algebraic multiplicity of the eigenvalue .

The eigenvalue of is called simple if its algebraic multiplicity is equal to one.

For each fixed the set of all complex number such that is not invertible is called the spectrum of . It is known that the spectrum of is an enumerable set of its eigenvalues (see [3, Theorem 5.2.1]). Moreover, there are constants such that is invertible for all and all in the set

(see [3, Theorem 3.6.1]).

To receive asymptotic formulas of the solutions with the coefficients regular with respect to the variable we require later that eigenvalues and eigenvectors of the pencil satisfy the following assumption.

Let be nonnegative integers, and let be real numbers such that . We say that the assumption for numbers is fulfilled if the following conditions are satisfied.

(i) The lines do not contain eigenvalues of the pencil , and the strip contains the eigenvalues , with the geometric multiplicities and the partial multiplicities , , not depending on . These eigenvalues are smooth functions on .(ii) A canonical system

of Jordan chains of corresponding to the eigenvalue () can be chosen, which consists of functions that are smooth for for all .

Theorem 4. Let be a positive integer. Let be real numbers such that , and , are not integers in the case . Suppose that the assumptions of Theorem 3 and the assumption for , , are fulfilled. Let be the generalized solution of the problem (8)-(9). Suppose further that the operators and are - admissible near the origin . Then,

(i) for the case , the solution admits the decomposition

where , are functions satisfying , for ,(ii) for the case , we assume further that if is integer for some and some , then for all ; then the solution admits the decomposition

where are functions satisfying , , for , and is the integer with , if is not an eigenvalue of ; otherwise is the maximal partial multiplicity of the eigenvalue .

4. Proofs of Theorems 2 and 3

First, let us introduce some more notations. For functions defined in and , we set

For functions and defined in , we set

Here and hereafter, we use instead of for the shortness.

To prove Theorem 2, it is needed to introduce the Gronwall-Bellman and interpolation inequalities stated in the following lemmas.

Lemma 5 (see [14, Lemma 3.1]). Assume are real-valued continuous on an interval , is nonnegative and integrable on and is nondecreasing satisfying

Then

Lemma 6 (see [15] ). For each positive real number and each integer , , there exists a positive real number which is dependent only on , , and such that the inequality

holds for all .

Proof of Theorem 2. 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 in that theorem can be omitted.

Let be a basis of which is orthonormal in . Put

where are the solution of the system of the following ordinary differential equations of second order:

with the initial conditions

Let us multiply (47) by . Take the sum with respect to from to to receive

Now adding this equality to its complex conjugate with noting that

by the formally self-adjointness of the operator , then integrating the obtained equality with respect to from to with using the integration by parts and (48), we arrive at

Noting that

we have from (51) that

Now we give estimations for the terms of (53). Firstly, by (2), we see that the left-hand side of (53) is greater than

We write as the sum of the two following terms:

Put

Then, by the Cauchy inequality, we have

By the Cauchy inequality and the interpolation inequality (45), for an arbitrary positive number , we have

where is a nonnegative constant independent of , , and . Now using again the Cauchy and interpolation inequalities, for an arbitrary positive number with , it holds that

where is a nonnegative constant independent of , , and . Also by Cauchy inequality we have

Now, to deal with the last term of (53), let us consider first the case . In this case, we use the following inequality:

where arbitrary. Combining the above estimations we get from (53) that

Now fix , and consider the function

We see that

We see that the function has a unique minimum at

Let us denote

Now we take real numbers , arbitrarily satisfying . Then there are positive real numbers , (), (), and such that

From now to the end of the present proof, we fix such constants , , , and and denote by the left-hand side of (62). It follows from (62) and (67) that

By the Gronwall-Bellman inequality (44), we deduce from (68) that

for all . Here we used the fact that, for ,

In the case of , instead of (61), we give the following inequality

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 .

Now multiplying both sides of this inequality by , then integrating them with respect to from to , we arrive at

where we used the notation

It is clear that is a norm in which is equivalent to the norm . Thus, it follows from (72) that

where is a constant independent of and .

From the inequality (74), by standard weak convergence arguments (see, e.g., [16, Ch. 7]), we can conclude that the sequence possesses a subsequence convergent to a function 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 Theorem 3. 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 in [7] can be removed.

Lemma 7. Suppose that all the assumptions of Theorem 3 are fulfilled. Then for and

where is a constant independent of and .

From the proof of [12, Lemma 5.3] we have the following lemma.

Lemma 8. Let , , , and . Suppose that is a solution of the following problem:

Then and

where the constant is independent of , and .

Proof of Theorem 3. According to Lemma 7 we have

Moreover, as in proof of Theorem 4.1 of [7], we have

Now we prove the theorem by induction on . By (78), . Thus, from (21) we have

for all and a.e. . Since for a.e. , by Lemma 8, we get from (80) that for a.e. and

where is a constant independent of , and . Now multiplying both sides of (81) with , then integrating with respect to from to and using estimates from Lemma 7, we obtain

where is a constant independent of and . Hence, the theorem is valid for .

Assume that the theorem is true for some nonnegative . We will prove it for . Differentiating times both sides of (80) with respect to we have

for all , a.e. . From (6) it follows that

for all and . Thus, from (83) we deduce

for all and a.e. , where

By the induction assumption, it holds that

Moreover,

by the assumption of the theorem, and

by Lemma 7. Thus, for a.e. , we have , , and

where is the constant independent of , and . Now we can repeat the arguments above to conclude that with the estimate (23) for . The proof is completed.

5. The Proof of Theorem 4

Let us first give some auxiliary lemmas.

Lemma 9. Let be a nonnegative integer, , and let , be real numbers, . Suppose that the assumption is fulfilled for the numbers . Let , let be real numbers, and let and be functions satisfying , for . Suppose that is a solution of the problem

Then admits the following representation:

where and are functions satisfying , for .

Proof. From the proof of Lemma 4.5 of [13] (see also [17, Lemma 4.1]), it is known that, for each fixed , the solution admits the representation (92) and the following inequality

holds for all , , where is a constant independent of , , , and . Now multiplying both sides of (93) by , then integrating with respect to from to , we see that , . The lemma is proved.

By applying Lemma 9 and repeating the arguments in the proof of [13, ], we get the following lemma.

Lemma 10. Suppose that all assumptions of Lemma 9 are fulfilled. Suppose further that the operators and are -admissible near the origin , where . Let be a solution of the problem

Then admits a representation of the form (92).

The following lemma follows directly from the proof of [13, Lemma 4.7].

Lemma 11. Let , and let be real numbers. Let

where and are given functions defined on , , respectively, satisfying ,, , . Suppose that if is an eigenvalue of for some , then it is an eigenvalue of for all with the geometric multiplicity and the partial multiplicities not depending on . Then there exists a solution of the problem (91) which has the form

where are functions defined on satisfying . Here if is not an eigenvalue of ; otherwise is the maximal partial multiplicity of .

Proof of Theorem 4. According to Theorem 3, we have

for .

Rewrite (8) in the form

Since by (75) and by the assumption, we have

for .

Now the assertions of the theorem are obtained by applying Lemma 10 and repeating almost word for word the proof of [13, Theorem 4.8].

Acknowledgment

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 101.01-2011.30.

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