Visco-elastic relaxation with a van der waals type stress

Contemporary science models certain phase transitions as conservation laws with a non-monotone stress-strain relation. The system of conservation laws are often hyperbolic except in a region in phase space, where the system is elliptic or of composite type. It is possible to prove existence and uniqueness of solutions for specific models which exhibit such change of type, but notions of admissibility and the connections between the mathematical theory and engineering experiments are not fully understood. In this paper we model phase transition by a visco-elastic relaxation law, and describe numerical experiments on this system. In particular we examine special initial data which, in the context of the mixed system, gives rise to multiple solutions.