On a shock problem involving a nonlinear viscoelastic bar

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 , .