- 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
© The Author(s) 2009
- Received: 3 July 2009
- Accepted: 8 December 2009
- Published: 12 January 2010
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.
- Generalize Solution
- Differential Operator
- Elliptic Equation
- Dirichlet Problem
- Hyperbolic Equation
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.
It is known that , then (see [7, Lemma ]).
and by , the spaces consisting of all functions such that generalized derivatives exist and belong to , (see ), with norms
For shortness, we set
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:
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:
Suppose that the following hypotheses are satisfied:
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.
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. .
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. ). 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 , we have the following lemma.
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  it follows that . Furthermore, (4.2) holds.
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.
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).
Now we will prove the global regularity of the solution.
The proof is complete.
Thus, the change of variables
has only real roots (see [10, Theorem , page 94]). Therefore, they are
This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam.
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