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

Nguyen Manh Hung,Nguyen Thanh Anh

doi:10.1155/2013/801314

Nguyen Manh Hung, Nguyen Thanh Anh

Open Access

https://doi.org/10.1155/2013/801314

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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.

Highlights

A large number of investigations have been devoted to boundary value problems in nonsmooth domains with conical points
[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 [6,7,8,9], 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

Summary

Introduction

A large number of investigations have been devoted to boundary value problems in nonsmooth domains with conical points. In [6,7,8,9], 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. 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].

Notations and the Formulation of the Problem

Statements of the Main Results

Proofs of Theorems 2 and 3

The Proof of Theorem 4