Qualitative Behavior of the Two-Phase Entropy Solution of a Forward-Backward Parabolic Problem

We discuss some qualitative aspects of a forward-backward parabolic problem that has been introduced in [L. C. Evans and M. Portilheiro, Math. Models Methods Appl. Sci., 14 (2004), pp. 1599–1620], [C. Mascia, A. Terracina, and A. Tesei, Evolution of stable phases in forward-backward parabolic equations, in Asymptotic Analysis and Singularities, Mathematical Society of Japan, Tokyo, 2007, pp. 451–478] and further analyzed in [C. Mascia, A. Terracina, and A. Tesei, Arch. Ration. Mech. Anal., 194 (2009), pp. 887–925]. This problem arises in models of phase transition in which two stable phases are separated by an interface. In particular, we consider here the problem of the extension in time of the solution constructed in [C. Mascia, A. Terracina, and A. Tesei, Arch. Ration. Mech. Anal., 194 (2009), pp. 887–925]. We analyze the regularity of the solution u defined in a domain R×(0,T) and give an estimate, depending on the initial datum, of the number of convex regions of the function u(·,t) for every t∈(0,T).