Finite time blow-up for a class of parabolic or pseudo-parabolic equations

Abstract

In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations: ut−aΔut−Δu+bu=k(t)|u|p−2u,(x,t)∈Ω×(0,T),where a≥0, b>−ł1 with ł1 being the principal eigenvalue for −Δ on H01(Ω) and k(t)>0. By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) J(u0;0)<0; (ii) J(u0;0)≤d(∞), where d(∞) is a nonnegative constant; (iii) 0<J(u0;0)≤Cρ(0), where ρ(0) involves the L2-norm or H01-norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level.