Communications in Mathematical Sciences

Volume 17 (2019)

Number 7

Existence of solutions to an anisotropic degenerate Cahn–Hilliard-type equation

Pages: 2035 – 2054

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n7.a11

Author

Marion Dziwnik (Department of Mathematics, University of Hamburg, Germany)

Abstract

We prove existence of solutions to an anisotropic Cahn–Hilliard-type equation with degenerate diffusional mobility. In particular, the mobility vanishes at the pure phases, which is typically used to model motion by surface diffusion. The main difficulty of the present existence result is the strong non-linearity given by the fourth-order anisotropic operator. Imposing particular assumptions on the domain and assuming that the strength of the anisotropy is sufficiently small enables to establish appropriate bounds which allow to pass to the limit in the regularized problem. In addition to the existence we show that the absolute value of the corresponding solutions is bounded by $1$.

Keywords

Cahn–Hilliard equation, degenerate mobility, anisotropic parabolic equations, existence/boundedness of solutions

2010 Mathematics Subject Classification

35K55, 35K65, 49Jxx, 74Gxx, 74Hxx, 82C26

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Copyright © 2019 by Marion Dziwnik.

Received 20 October 2016

Accepted 10 July 2019

Published 6 January 2020