Analytical and numerical dissipativity for the space-fractional Allen-Cahn equation

This paper is concerned with the analytical and numerical dissipativity of the space-fractional Allen-Cahn equation, a generalization of the classic Allen-Cahn equation by replacing the local Laplacian with a nonlocal fractional Laplacian. It is first proved that the continuous dynamical system is dissipative as its local counterpart in H alpha and Lq, q = 2k + 2 for k >= 0, spaces. Then it is shown that the backward Euler method preserves the dissipativity of the underlying system, that is, the discrete-in-time dynamical system with time-step parameter Tau is still dissipative in H alpha and Lq spaces. The existence of the global attractor for both continuous and discrete dynamical systems are then obtained. A numerical example is given to confirm the theoretical results.(c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.