The initial-value problem for the dissipative equation with the extra-force-term is considered. The dissipative effect on this equation is given by the fractional Laplacian which describes the anomalous diffusion. The main goal of this paper is to give an estimate on the difference between solutions and their asymptotic expansion as the space and the time variables tend to infinity. Furthermore, as an application, the large-time behavior of solutions to some concrete problems are discussed. (C) 2015 Elsevier Inc. All rights reserved.