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

Abstract

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 Hα 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 τ is still dissipative in Hα 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.