$L_p$-theory for a Cahn-Hilliard-Gurtin system

Mathias Wilke

doi:10.3934/eect.2012.1.393

Article Contents

2012, Volume 1, Issue 2: 393-429. Doi: 10.3934/eect.2012.1.393

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$L_p$-theory for a Cahn-Hilliard-Gurtin system

Mathias Wilke^1,

1.
Institut für Mathematik, Martin-Luther Universität Halle-Wittenberg, 06099 Halle

Received: February 29, 2012

Revised: May 31, 2012

Published: November 2012

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Abstract

In this paper we study a generalized Cahn-Hilliard equation which was proposed by Gurtin [9]. We prove the existence and uniqueness of a local-in-time solution for a quasilinear version, that is, if the coefficients depend on the solution and its gradient. Moreover we show that local solutions to the corresponding semilinear problem exist globally as long as the physical potential satisfies certain growth conditions. Finally we study the long-time behaviour of the solutions and show that each solution converges to a equilibrium as time tends to infinity.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35B38, 35B40, 35B65, 82C26.

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