Long-time behaviour of two-phase solutions to a class of forward-backward parabolic equations

We consider two-phase solutions to the Neumann initial-boundary value problem for the parabolic equation ut = [φ(u)]xx , where φ is a nonmonotone cubic-like function. First, we prove global existence for a restricted class of initial data u0, showing that two-phase solutions can be obtained as limiting points of the family of solutions to the Neumann initial-boundary value problem for the regularized equation ut = [φ(u )]xx + eutxx (e > 0). Then, assuming global existence, we study the long-time behaviour of two-phase solutions for any initial datum u0.