TY - JOUR

T1 - Generation of interface for solutions of the mass conserved Allen-Cahn equation

AU - Hilhorst, Danielle

AU - Matano, Hiroshi

AU - Nguyen, Thanh Nam

AU - Weber, Hendrik

N1 - Funding Information:
The research of the second author was partially supported by KAKENHI (16H02151). The research of the first, second, and third authors was supported by the GDRI ReaDiNet. The research of the third author was supported by ERC Adora and by Plan Cancer HTE program EcoAML. The research of the fourth author was supported by EPSRC First Grant EP/L018969/1 and by the Royal Society through the University Research Fellowship UF140187.
Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/12/31

Y1 - 2020/12/31

N2 - 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.

AB - 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.

KW - Absence of interface

KW - Allen-Cahn

KW - Generation of interface

KW - Nonlinear PDE

KW - Reaction-diffusion equation

KW - Singular perturbation

UR - http://www.scopus.com/inward/record.url?scp=85091331645&partnerID=8YFLogxK

U2 - 10.1137/18M1204747

DO - 10.1137/18M1204747

M3 - Article

AN - SCOPUS:85091331645

SN - 0036-1410

VL - 52

SP - 2624

EP - 2654

JO - Siam Journal on Mathematical Analysis

JF - Siam Journal on Mathematical Analysis

IS - 3

ER -