Communications in Mathematical Sciences
Volume 17 (2019)
Existence of solutions to an anisotropic degenerate Cahn–Hilliard-type equation
Pages: 2035 – 2054
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 $1$.
Cahn–Hilliard equation, degenerate mobility, anisotropic parabolic equations, existence/boundedness of solutions
2010 Mathematics Subject Classification
35K55, 35K65, 49Jxx, 74Gxx, 74Hxx, 82C26
Copyright © 2019 by Marion Dziwnik.
Received 20 October 2016
Accepted 10 July 2019
Published 6 January 2020