Abstract: Evolution of a phase transition system is an importanttopic with many applications. The crucial difficulty lies in the instabilityof ellipticity. In this paper, we briefly discuss two types ofapproaches to resolve this difficulty, i.e. to employ generalized entropyconditions, or to include dissipation mechanisms. For the latter one, wereview some recent progresses for systems with both high-order andlow-order dissipations. Theoretical and numericalinvestigations reveal the mechanism and the complex pattern evolving from thenonlinear interaction between dissipations and ellipticity.