Stability of Traveling Waves with Degenerate Shock for System of One-Dimensional Viscoelastic Model

The stability of traveling wave solutions with shock profile for a system for a one-dimensional viscoelastic material model is investigated. The stress function is not necessarily assumed to be convex or concave, so that the shock condition may be degenerate. Ln some restricted conditions on the stress function it is shown that the traveling wave solution is asymptotically stable, provided the shock is weak and the initial disturbance is small and of integral zero. Proof is given by applying an elementary weighted energy method to the integrated system of the original one. The weight function which is suitably selected plays a key role in the procedure.