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

TL;DR

This paper establishes local and global existence, uniqueness, and long-term convergence of solutions for a generalized Cahn-Hilliard-Gurtin system, extending the mathematical understanding of this phase separation model.

Contribution

It proves the existence, uniqueness, and long-time behavior of solutions for a quasilinear Cahn-Hilliard-Gurtin system with solution-dependent coefficients.

Findings

Local-in-time existence and uniqueness of solutions

Global existence under growth conditions on the potential

Convergence of solutions to equilibrium over time

Abstract

In this paper we study a generalized Cahn-Hilliard equation which was proposed by Gurtin. 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.