Two-phase entropy solutions of forward-backward parabolic problems with unstable phase

In this paper we study a two-phase problem for a forward-backward parabolic equation with diffusion function of cubic type. Existence and uniqueness for these kind of problems were obtained in literature in the case in which the phases are both stable. Here we consider the situation in which the unstable phase is taken in account, obtaining not trivial solutions of the problem. It is interesting to note that such solutions are given by solving generalized Abel's equations.