Existence of solutions to an anisotropic degenerate Cahn--Hilliard-type equation
- Dziwnik, Marion
- Jachalski, Sebastian
2010 Mathematics Subject Classification
- 74Gxx 74Hxx 35K55 35K65 49Jxx 82C26
- degenerate Cahn--Hilliard equation, anisotropic parabolic equations, existence of solutions, boundedness of solutions
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 auxiliary results which play an essential part in the present existence proof. In addition to the existence we show that the absolute value of the corresponding solutions is bounded by 1.