Open Access

On a shock problem involving a nonlinear viscoelastic bar

  • Nguyen Thanh Long1Email author,
  • Alain Pham Ngoc Dinh2 and
  • Tran Ngoc Diem1
Boundary Value Problems20052005:718156

DOI: 10.1155/BVP.2005.337

Received: 3 August 2004

Published: 13 November 2005

Abstract

We treat an initial boundary value problem for a nonlinear wave equation in the domain , . The boundary condition at the boundary point of the domain for a solution involves a time convolution term of the boundary value of at , whereas the boundary condition at the other boundary point is of the form with and given nonnegative constants. We prove existence of a unique solution of such a problem in classical Sobolev spaces. The proof is based on a Galerkin-type approximation, various energy estimates, and compactness arguments. In the case of , the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution of this problem up to order in two small parameters , .

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, University of Natural Science
(2)
Laboratoire de Mathématiques et Applications, physique Mathématique d'Orléans (MAPMO), UMR 6628, Bâtiment de Mathématiques, Université d'Orléans

Copyright

© Long et al. 2005