Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory

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April 2010, 26(2): 625-647. doi: 10.3934/dcds.2010.26.625

Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory

Jan Prüss ^1, , Vicente Vergara ^2, and Rico Zacher ^3,

1.	Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik, Theodor-Lieser-Strasse 5, D-06120 Halle, Germany
2.	Universidad de Tarapacá, Instituto de Alta Investigación, Antofagasta N. 1520, Arica, Chile
3.	Fachbereich Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle

Received January 2009 Revised April 2009 Published October 2009

Abstract
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In this paper we study a temperature dependent phase field model with memory. The case where both the equation for the temperature and that for the order parameter is of fractional time order is covered. Under physically reasonable conditions on the nonlinearities we prove global well-posedness in the $L_p$ setting and show that each solution converges to a steady state as time goes to infinity.

Keywords: Cahn-Hilliard equation with memory, integro-differential equations, phase field system with memory, Łojasiewicz-Simon inequality, Maximal regularity, convergence to equilibrium..

Mathematics Subject Classification: Primary: 45K05; Secondary: 45M0.

Citation: Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625

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