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April 2013, 33(4): 1657-1697. doi: 10.3934/dcds.2013.33.1657
Sobolev approximation for two-phase solutions of forward-backward parabolic problems
Flavia Smarrazzo 1, and Andrea Terracina 1,
Dipartimento di Matematica "G. Castelnuovo,", Sapienza Università di Roma, Piazzale A. Moro 5, 00185 Roma, Italy, Italy
Received September 2011 Revised July 2012 Published October 2012
We discuss some properties of a forward-backward parabolic problem that 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.
Keywords: Forward-backward parabolic equations, a priori estimates, uniqueness results., two-phase solutions, pseudo parabolic regularization.
Mathematics Subject Classification: Primary: 35K20, 35D99; Secondary: 35K7.
Citation: Flavia Smarrazzo, Andrea Terracina. Sobolev approximation for two-phase solutions of forward-backward parabolic problems. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1657-1697. doi: 10.3934/dcds.2013.33.1657
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Flavia Smarrazzo Andrea Terracina