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 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. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This study investigates the dynamics of a nonlocal Allen-Cahn equation with Neumann boundary conditions and determines the stability/instability of all symmetric solutions and a part of asymmetric solutions, with numerical simulations supporting the results.