Stability of solutions of a class of nonlinear fractional Laplacian parabolic problems

Abstract

In this paper we study the forward and the backward in time problems for a class of nonlinear partial differential equations of fractional Laplace operator with nonlinear source function. The equations are generalized from lots of well-known equations such as the Burger equation, the Cahn–Hilliard equation. Herein we investigate the stability of the solution of the forward problem depended on the fractional order of the operator and the instability of the backward problem, and then, we establish a regularization method. Some numerical experiments are performed to verify efficiency and accuracy of our method.