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.