Asymptotic behaviour of solutions to some pseudoparabolic equations

The aim of this paper is to investigate the behavior as t ! 1 of solutions to the Cauchy problem u t ? u t ? u ? (b; ru) = r F (u); u(x; 0) = u 0 (x), where > 0 is a xed constant, t 0, x 2 IR n. First, we prove that if u is the solution to the linearized equation, i.e with rF(u) 0, then u decays like a solution for the analogous problem to the heat equation. Moreover, the long time behavior of u is described by the heat kernel. Next, analogous results are established for the nonlinear equation with some assumptions imposed on F , p, and the initial condition u 0 .