An interface problem: The two-layer shallow water equations
Singular limits for the two-phase Stefan problem
November 2013, 33(11&12): 5347-5377. doi: 10.3934/dcds.2013.33.5347
Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation
Michel Pierre 1, and Morgan Pierre 2,
ENS Cachan Bretagne, IRMAR, EUB, Campus de Ker Lann, 35170 Bruz
Université de Poitiers, Laboratoire de Mathématiques, et Applications UMR CNRS 7348, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France
Received March 2012 Revised December 2012 Published May 2013
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: singular measures., Pseudo-parabolic, capacity, global existence, monotone operator, Allen-Cahn, blow-up.
Mathematics Subject Classification: Primary: 35K85, 35D99; Secondary: 65M6.
Citation: Michel Pierre, Morgan Pierre. Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5347-5377. doi: 10.3934/dcds.2013.33.5347
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Michel Pierre Morgan Pierre