Global existence and long-time behaviour for a singular integro-differential phase-field system

Elena Bonetti; Elisabetta Rocca

doi:10.3934/cpaa.2007.6.367

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2007, Volume 6, Issue 2: 367-387. Doi: 10.3934/cpaa.2007.6.367

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Global existence and long-time behaviour for a singular integro-differential phase-field system

Elena Bonetti^1, and
Elisabetta Rocca^2,

1.
Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, via Ferrata, 1, 27100, Pavia
2.
Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via Saldini, 50, 20133, Milano

Received: January 31, 2006

Revised: July 31, 2006

Published: May 2007

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Abstract

This paper deals with a singular integro-differential PDE system describing phase transitions in terms of nonlinear evolution equations for micromotions and for the entropy. The model is derived from a non-convex free energy functional, possibly accounting for thermal memory effects. After recovering a global existence result for a related initial and boundary value problem, the long-time behaviour of the solutions is investigated. In particular, it is proved that the elements of the $\omega$-limit set (i.e. the cluster points) of the solution trajectories solve the steady state system which is naturally associated to the evolution problem.

Keywords:

Singular integro-differential phase-field system,
phase transitions,
long-time behaviour of solutions,
$\omega$-limit.

Mathematics Subject Classification: 34B16, 45D05, 80A22, 74H40.

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