On a shock problem involving a nonlinear viscoelastic bar

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,.