ON A SHOCK PROBLEM INVOLVING

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