Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation

Michel Pierre; Morgan Pierre

doi:10.3934/dcds.2013.33.5347

Article Contents

2013, Volume 33, Issue 11&12: 5347-5377. Doi: 10.3934/dcds.2013.33.5347

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Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation

Michel Pierre^1, and
Morgan Pierre^2,

1.
ENS Cachan Bretagne, IRMAR, EUB, Campus de Ker Lann, 35170 Bruz
2.
Université de Poitiers, Laboratoire de Mathématiques, et Applications UMR CNRS 7348, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil

Received: February 29, 2012

Revised: November 30, 2012

Published: October 2013

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Abstract

The main goal of this paper is to prove existence of global solutions in time for an Allen-Cahn-Gurtin model of pseudo-parabolic type. Local solutions were known to ``blow up" in some sense in finite time. It is proved that the equation is actually governed by a monotone-like operator. It turns out to be multivalued and measure-valued. The measures are singular with respect to the Lebesgue measure. This operator allows to extend the local solutions globally in time and to fully solve the evolution problem. The asymptotic behavior is also analyzed.

Keywords:

Mathematics Subject Classification: Primary: 35K85, 35D99; Secondary: 65M60.

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