Generation of interface for solutions of the mass conserved Allen-Cahn equation

Danielle Hilhorst, Hiroshi Matano, Thanh Nam Nguyen, Hendrik Weber

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we study the generation of interface for the solution of the mass conserved Allen-Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initial functions that are independent of ϵ, the solution generally develops a steep transition layer of thickness O(ϵγ) (0 < γ ≤ 1) at a certain time of order ϵ2 |ln ϵ|. In some cases, we prove that the thickness of the interface is exactly of order ϵ, which is the optimal thickness estimate. We note that the comparison principle does not hold for our equation because of the nonlocal term so that the methods that were employed in the earlier studies of the standard Allen-Cahn equation do not work. We will therefore take a different approach, which is based on the fine analysis of the long-time behavior of the corresponding nonlocal ODEs and some energy estimates.

Original languageEnglish
Pages (from-to)2624-2654
Number of pages31
JournalSiam Journal on Mathematical Analysis
Volume52
Issue number3
DOIs
Publication statusPublished - 31 Dec 2020

Keywords

  • Absence of interface
  • Allen-Cahn
  • Generation of interface
  • Nonlinear PDE
  • Reaction-diffusion equation
  • Singular perturbation

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this