Sharp interface motion of Allen-Cahn equation with non-linear diffusion

2024/06/04 Tue 16:45 - 18:15
Hyunjoon Park
Meiji University

Allen-Cahn equation have been studied in order to understand the phase transition phenomena in various situation. In this talk we aim to understand the movement of the sharp interface when the diffusivity has non-linear structure, where such structure is natural in modeling the movement of the interacting particle. We begin the talk by considering the case when the non-linear diffusion is non-degenerate and see how such property affects the motion of the interface. Then, we study the degenerate case by considering the porous medium diffusion; where the non-linear diffusion becomes u^m,m>1. Unlike the non-degenerate diffusion case, porous medium diffusion gives an asymmetric feature to the stable steady states which may yield a difference between the motion of the interface and the free boundary of the solution. The aim of the talk is to explain the motion of the interface and the difference by constructing sub- and super-solutions of the Allen-Cahn equation.