Stability for stationary solutions of a nonlocal Allen-Cahn equation

We consider the dynamics of a nonlocal Allen-Cahn equation with Neumann boundary conditions on an interval. Our previous papers [2,3] obtained the global bifurcation diagram of stationary solutions, which includes the secondary bifurcation from the odd symmetric solution due to the symmetric breakin...

Full description

Saved in:
Bibliographic Details
Published in:Journal of Differential Equations 2021-02, Vol.275, p.581-597
Main Authors: Miyamoto, Yasuhito, Mori, Tatsuki, Tsujikawa, Tohru, Yotsutani, Shoji
Format: Article
Language:English
Subjects:
Allen-Cahn equation
Bifurcation
Nonlocal term
Stability
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We consider the dynamics of a nonlocal Allen-Cahn equation with Neumann boundary conditions on an interval. Our previous papers [2,3] obtained the global bifurcation diagram of stationary solutions, which includes the secondary bifurcation from the odd symmetric solution due to the symmetric breaking effect. This paper derives the stability/instability of all symmetric solutions and instability of a part of asymmetric solutions. To do so, we use the exact representation of symmetric solutions and show the distribution of eigenvalues of the linearized eigenvalue problem around these solutions. And we show the instability of asymmetric solutions by the SLEP method. Finally, our results with respect to stability are supported by some numerical simulations.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2020.11.024