ASYMPTOTIC STABILITY OF VISCOUS SHOCK PROFILE FOR NON-CONVEX SYSTEM OF ONE-DIMENSIONAL VISCOELASTIC MATERIALS WITH BOUNDARY EFFECT

This paper is concerned with the asymptotic behavior of solution to the initial-boundary value problem on the half space R+ for a one-dimensional non-convex system of viscoelastic materials. The initial data has constant state at infinity and the velocity is imposed zero at the boundary x = 0. By virture of the boundary effect, the solution is expected to behave as outgoing viscous shock profile. When the initial data is suitably close to the corresponding outgoing viscous shock profile which is suitably away from the boundary, it is proved that the unique global solution exists in time and tends toward the properly shifted shock profile as the time goes to infinity. The result is given by a weighted energy method.