Global well-posedness in  uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$

Sergey Zelik; Jon Pennant

doi:10.3934/cpaa.2013.12.461

Article Contents

2013, Volume 12, Issue 1: 461-480. Doi: 10.3934/cpaa.2013.12.461

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Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$

Sergey Zelik^1, and
Jon Pennant^2,

1.
Department of Mathematics, University of Surrey, Guildford, GU2 7XH
2.
University of Surrey, Guildford, GU2 7XH

Received: June 30, 2011

Revised: April 30, 2012

Published: December 2012

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Abstract

We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation where, in addition, the dissipativity of the associated solution semigroup is established.

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Mathematics Subject Classification: 35B41, 35L05, 74K15.

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