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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.