On initial boundary value problem with Dirichlet integralconditions for a hyperbolic equation with the Besseloperator

Abdelfatah Bouziani

doi:10.1155/s1110757x03204034

Abstract

We consider a mixed problem with Dirichlet and integral conditions for a second‐order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

Highlights

In the recent years, hyperbolic equations with integral condition(s) have received considerable attention
Many processes in porous media can be described by secondorder hyperbolic equations with an integral condition [14, 15]
The presence of an integral term in boundary conditions can greatly complicate the application of standard functional or numerical methods, owing to the fact that the elliptic differential operator with integral condition is no longer positive definite in the usual function spaces, which poses the main source of difficulty

Summary

Introduction

Hyperbolic equations with integral condition(s) have received considerable attention. We are concerned with a boundary value problem with an integral condition for a second-order hyperbolic equation with the Bessel operator. It can be a part in the contribution of the development of the energy-integral method for solving such problems. In this paper, following the method presented, for instance, in [6], we prove that problem (1.1) possesses a unique strongly generalized solution, in weighted Sobolev spaces, that depends continuously on the right-hand side of (1.1a), the initial conditions (1.1b) and (1.1c), and the boundary conditions (1.1d) and (1.1e).

Uniqueness and continuous dependence

Conclusion