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

Flavia Smarrazzo; Andrea Terracina

doi:10.3934/dcds.2013.33.1657

Article Contents

2013, Volume 33, Issue 4: 1657-1697. Doi: 10.3934/dcds.2013.33.1657

This issue Previous Article Non-integrability of generalized Yang-Mills Hamiltonian system Next Article Well-posedness for a modified two-component Camassa-Holm system in critical spaces

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

Flavia Smarrazzo^1, and
Andrea Terracina^1,

1.
Dipartimento di Matematica "G. Castelnuovo,", Sapienza Università di Roma, Piazzale A. Moro 5, 00185 Roma

Received: August 31, 2011

Revised: June 30, 2012

Published: March 2013

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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.

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Mathematics Subject Classification: Primary: 35K20, 35D99; Secondary: 35K70.

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