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

Jan Prüss; Vicente Vergara; Rico Zacher

doi:10.3934/dcds.2010.26.625

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2010, Volume 26, Issue 2: 625-647. Doi: 10.3934/dcds.2010.26.625

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

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

Received: December 31, 2008

Revised: March 31, 2009

Published: June 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 45K05; Secondary: 45M05.

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