Asymptotic Behaviour of Solutions to some Pseudoparabolic Equations

The aim of this paper is to investigate the behaviour as t→∞ of solutions to the Cauchy problem u_t?Δu_t?νΔu?(b, ?u)=??F(u), u(x, 0)=u₀(x), where ν>0 is a fixed constant, t?0, x∈?ⁿ. First, we prove that if u is the solution to the linearized equation, i.e. with ??F(u)≡0, then u decays like a solution for the analogous problem to the heat equation. Moreover, the long-time behaviour of u is described by the heat kernel. Next, analogous results are established for the non-linear equation with some assumptions imposed on F, p, and the initial condition u₀. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.