The stability of the equilibria of the Allen–Cahn equation with fractional diffusion
Hongmei Cheng, Rong Yuan · Applicable Analysis · 2017
This paper is mainly concerned with the stability of the equilibria of the Allen–Cahn equation where the standard Laplacian diffusion is replaced by the fractional Laplacian diffusion. We shall show that the state u≡0 is stable with respect to some perturbations which are not too large on a set, but is unstable with respect to some perturbations with bounded support set. We use the comparison principle to obtain the results.