- Research Article
- Open Access

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

- NguyenManh Hung
^{1}and - VuTrong Luong
^{2}Email author

**Received:**3 July 2009**Accepted:**8 December 2009**Published:**12 January 2010

## Abstract

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.

## Keywords

- Generalize Solution
- Differential Operator
- Elliptic Equation
- Dirichlet Problem
- Hyperbolic Equation

## 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 us [1–4], 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.

In [5, 6] we showed the existence of a sequence of smooth domains such that and . Furthermore, we proved the existence, the uniqueness, and the smoothness with respect to time variable of the generalized solution by approximating boundary method, which can be applied for nonlinear equations. With the help of the results in [5, 6] as well as the results for elliptic boundary value problems in [7, 8], we can deal with the regularity with respect to both time variables and spatial ones of the solution.

Our paper is organized as follows: in Section 2, we introduce exterior cusp domain and weight Sobolev spaces. In Section 3, we will state the formulation of the problem. The main results, Theorems 4.3, 4.6, and 4.7, are stated in Section 4, and examples are given in Section 5.

## 2. Cusp Domain and Weighted Sobolev Spaces

Let be an infinitely differentiable positive function on the interval satisfying the following conditions:

(i) for

(ii) .

takes the set onto the half-cylinder . Moreover, it follows that

We extend the functions to an infinitely differentiable positive function on the interval The space can be defined as the closure of the set with respect to the norm

It is known that , then (see [7, Lemma ]).

We also denote by the Sobolev space of functions and that have generalized derivatives , . The norm in this space is defined as follows:

The space is the completion of in norm of the space .

Set ; we proceed to introduce some functional spaces. Let be Banach spaces, we denote by the spaces consisting of all measurable functions with norm

and by , the spaces consisting of all functions such that generalized derivatives exist and belong to , (see [9]), with norms

For shortness, we set

To simplify notation, we continue to write instead of .

## 3. Formulation of the Problem

Let us consider the partial differential operator of order

satisfy the condition of stabilization for for a.e. in (see [7, Section ]). Then the coefficients of the operators , which arise from operators via the coordinate change , stabilize for . If we replace the coefficients of the differential operator by their limits for , we get differential operator which has coefficients depending only on and (for the convenience in use, we denote also by ).

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

We also suppose that the form is -elliptic uniformly with respect to , that is, the inequality

is valid for all and all , where is the positive constant independent of and . In this paper, we consider the following problem:

where and are derivatives with respect to the outer unit normal of .

Definition 3.1.

holds for all .

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 were established in [5, 6] according to following theorems:

Theorem 3.2.

where is a constant independent of and .

Theorem 3.3.

Suppose that the following hypotheses are satisfied:

(i)

(ii)

(iii) .

where is a constant independent of and .

Owing to the support of the following proposition, we can apply the results of the Dirichlet problem for elliptic equation in domains with exterior cusps.

Proposition 3.4.

where .

Proof.

holds for all for all . It follows that is a generalized solution in of the Dirichlet problem for elliptic equation (3.13), for a.e. .

## 4. The Main Results

In this section, we would like to present the main results of the study which is based on our previous results (cf. [5, 6]) and the results of elliptic equations in cusp domains (cf. [7]). For the start of this section, we denote by the operator corresponding to the parameter-depending boundary value problem

For each we have the operator pencil to be Fredholm, and its spectrum consists of a countable number of isolated eigenvalues. Similarly to Theorem in [7], we have the following lemma.

Lemma 4.1.

where the constant is independent of .

Proof.

Let be the function which arises from via the coordinate change . We set ; then from the properties of the mapping (2.2) and from inequality (4.6), it follows that . Since is the solution of an elliptic equation in with coefficients which stabilize for , that is,

as if , we conclude that From Corollary in [7] it follows that . Furthermore, (4.2) holds.

Lemma 4.2.

where the constant is independent of .

Proof.

By integrating the inequality above with respect to from to , and using the estimates for derivatives of with respect to again, we obtain , which satisfies inequality (4.9).

Theorem 4.3.

where the constant is independent of .

Proof.

where is a constant independent of , and .

By using (4.15) and estimates for derivatives of with respect to in Theorem 3.3, we have

Remark 4.4.

Let be a sufficiently small positive number. Suppose that and the strip contains no eigenvalues of , ; then the generalized solution of problem (3.6)–(3.8), such that if , belongs to the . In fact, setting , we obtain the first initial boundary value problem which differs little from (3.6)–(3.8). Therefore, , and then . Using the remark above and Lemma 4.1, we obtain the following theorem.

Theorem 4.5.

where the constant is independent of .

This theorem is proved by arguments analogous to those proofs of Lemma 4.2 and Theorem 4.3. Next, we will prove the well regularity of the generalized solution of problem (3.6)–(3.8).

Theorem 4.6.

where the constant is independent of and .

Proof.

It implies that (4.19) holds for . The proof is complete.

Now we will prove the global regularity of the solution.

Theorem 4.7.

where the constant is independent of and .

Proof.

The proof is complete.

## 5. Examples

where , , , as and , and , .

where . It follows the results of Section 4 that if , where is the least positive root of the Bessel function , then problem (5.2) has a unique solution in and we have the estimate

Moreover, if , and for , then and satisfies

For the two-dimensional case , and letting , we consider problem (5.2) in the cylinder , where is a bounded domain in is smooth, and

Thus, the change of variables

transforms

With notations , we have

Clearly, coefficients of differential operator stabilize for , and the limit differential operator of (denoted by for convenience) is

We denote also by the operator corresponding to the parameter-depending boundary value problem

Eigenvalues of are roots of the Bessel function

has only real roots (see [10, Theorem , page 94]). Therefore, they are

It is easy to see that is the least positive root of the Bessel function . From arguments above in combination with Theorems 4.6 and 4.7, we obtain the following results:

Theorem 5.1.

In case that boundary of has some cuspidal points, then by arguments analogous to Section 4, we consequently obtain the similar results.

## Declarations

### Acknowledgment

This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam.

## Authors’ Affiliations

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