ASYMPTOTIC EXPANSION OF SOLUTIONS TO THE DISSIPATIVE EQUATION WITH FRACTIONAL LAPLACIAN

In this paper, the Cauchy problem for the linear dissipative equation with a potential is studied. The dissipative effect of this equation is given by the fractional Laplacian. When a dissipative equation is considered, the fractional Laplacian describes the anomalous diffusion. The main goal of this paper is to derive the large-time behavior of decaying solutions. Particularly, the estimate on the difference between solutions and their asymptotic expansion as t -> infinity is given. The spatial decay of this difference is also derived. Generally speaking, when a dissipative equation with the fractional Laplacian is studied, it is difficult to obtain the asymptotic expansion of solutions with high order. The anomalous diffusion causes this difficulty. The spatial decay of the difference between solutions and their asymptotic expansion provides the asymptotic expansion with arbitrary high order. Furthermore, as an application, the large-time behavior of solutions to a nonlinear problem is discussed.