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February  2009, 25(1): 63-81. doi: 10.3934/dcds.2009.25.63

Long time convergence for a class of variational phase-field models

Pierluigi Colli 1, , Danielle Hilhorst 2, , Françoise Issard-Roch 2, and  Giulio Schimperna 3,

1. 

Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100 Pavia
2. 

CNRS and Laboratoire de Mathématiques, Université Paris-Sud 11, Bat. 425, F-91405 Orsay, France
3. 

Dipartimento di Matematica "F.Casorati", Università di Pavia, Via Ferrata, 1, I-27100 Pavia

Received  December 2007 Published  June 2009

In this paper we analyze a class of phase field models for the dynamics of phase transitions which extend the well-known Caginalp and Penrose-Fife phase field models. We prove the existence and uniqueness of the solution of a corresponding initial boundary value problem and deduce further regularity of the solution by exploiting the so-called regularizing effect. Finally we study the long time behavior of the solution and show that it converges algebraically fast to a stationary solution as $t$ tends to infinity.
Keywords: Simon-Lojasiewicz inequality., gradient flow, Phase transition, $\omega$-limit set.
Mathematics Subject Classification: Primary: 35B40, 35K45; Secondary: 80A2.
Citation: Pierluigi Colli, Danielle Hilhorst, Françoise Issard-Roch, Giulio Schimperna. Long time convergence for a class of variational phase-field models. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 63-81. doi: 10.3934/dcds.2009.25.63
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