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.

Sobolev approximation for two-phase solutions of forward-backward parabolic problems / Flavia, Smarrazzo; Terracina, Andrea. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - STAMPA. - 33:4(2013), pp. 1657-1697. [10.3934/dcds.2013.33.1657]

Sobolev approximation for two-phase solutions of forward-backward parabolic problems

TERRACINA, Andrea
2013

Abstract

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.
2013
pseudo parabolic regularization; two-phase solutions; uniqueness results; forward-backward parabolic equations; a priori estimates
01 Pubblicazione su rivista::01a Articolo in rivista
Sobolev approximation for two-phase solutions of forward-backward parabolic problems / Flavia, Smarrazzo; Terracina, Andrea. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - STAMPA. - 33:4(2013), pp. 1657-1697. [10.3934/dcds.2013.33.1657]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/458997
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