Boundary Value Problems 
Volume 2005 (2005), Issue 3, Pages 337-358
doi:10.1155/BVP.2005.337

On a shock problem involving a nonlinear viscoelastic bar

Nguyen Thanh Long,1 Alain Pham Ngoc Dinh,2 and Tran Ngoc Diem1

 1Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University HoChiMinh City, 227 Nguyen Van Cu Street, Dist.5, HoChiMinh City, Vietnam
2Laboratoire de Mathématiques et Applications, physique Mathématique d'Orléans (MAPMO), UMR 6628, Bâtiment de Mathématiques, Université d'Orléans, Orléans Cedex 2 BP 6759, France
Received 3 August 2004; Revised 23 December 2004

Abstract

We treat an initial boundary value problem for a nonlinear wave equation utt−uxx+K|u|αu+λ|ut|βut=f(x,t) in the domain 0<x<1, 0<t<T. The boundary condition at the boundary point x=0 of the domain for a solution u involves a time convolution term of the boundary value of u at x=0, whereas the boundary condition at the other boundary point is of the form ux(1,t)+K1u(1,t)+λ1ut(1,t)=0 with K1 and λ1 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 α=β=0, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution (u,P) of this problem up to order N+1 in two small parameters K, λ.