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) = u o (x), where ν > 0 is a fixed constant, t ≥ 0, x ∈ R 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 u o .

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