On a shock problem involving a nonlinear viscoelastic bar

22 pages (To appear in Boundary Value Problem)We treat an initial boundary value problem for a nonlinear wave equation $u_{tt} - u_{xx} + K|u|^{\alpha} + \lambda|u_t|^{\beta}u_t = f(x,t)$ 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 at $x=0$ whereas the boundary condition at the boundary condition at the other boundary point is of the form $u_x(1,t) + K_1u(1,t) + \lambda_{1}u_t(1,t) = 0$ with $K_1$ and $\lambda_{1}$ given non-negative constants. We prove existence of a unique solution of such a problem in classical Sobolev. The proof is based on a Galerkin type approximation, various energy estimates. In the case of $\alpha=\beta=0$, the regularity of solution is studied. Finally, we obtain an asymptotic expansion of the solution (u,P) of this problem up to order N+1 in two small parameters $K, \lambda$