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

Abstract

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 _u_0, 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_ε = [ϕ(_u_ε)]xx + ε_utxx_ε (ε > 0). Then, assuming global existence, we study the long-time behaviour of two-phase solutions for any initial datum _u_0.