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
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
Copyright © 2005 Nguyen Thanh Long et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.