Sobolev approximation for two-phase solutions of forward-backward parabolic problems
Flavia Smarrazzo, Andrea Terracina · Discrete and Continuous Dynamical Systems · 2012
We discuss some properties of a forward-backward parabolic problemthat arises in models of phase transition in which two stable phases are separated by an interface.Here we consider a formulation of the problem that comes from a Sobolev approximation of it.In particular we prove uniqueness of the previous problem extending to nonlinear diffusion function a result obtained in [21] in the piecewise linear case. Moreover, we analyze the third order partial differential problem that approximates the forward-backward parabolic one. In particular, for some classes of initial data, we obtain a priori estimates that generalize that proved in [22]. Using these results we study the singular limit of the Sobolev approximation, as a consequence we obtain existence of the forward-backward problem for a class of initial data.