On a shock problem involving a nonlinear viscoelastic bar. (13th November 2005) - Long, Nguyen Thanh; Dinh, Alain Pham Ngoc; Diem, Tran Ngoc
Abstract : We treat an initial boundary value problem for a nonlinear wave equationu t t − u x x + K | u | α u + λ | u t | β u t = f ( x, t ) in the domain0 < x < 1, 0 < t < T . The boundary condition at the boundary pointx = 0 of the domain for a solutionu involves a time convolution term of the boundary value ofu atx = 0, whereas the boundary condition at the other boundary point is of the formu x ( 1, t ) + K 1 u ( 1, t ) + λ 1 u t ( 1, t ) = 0 withK 1 andλ 1 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 ofα = 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 orderN + 1 in two small parametersK, λ .