Finite time blow-up for a class of parabolic or pseudo-parabolic equations. (15th May 2018) - Sun, Fenglong; Liu, Lishan; Wu, Yonghong
Abstract: In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations: u t − a Δ u t − Δ u + b u = k ( t ) | u | p − 2 u, ( x, t ) ∈ Ω × ( 0, T ), where a ≥ 0, b > − ł 1 with ł 1 being the principal eigenvalue for − Δ on H 0 1 ( 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 ( u 0 ; 0 ) < 0 ; (ii) J ( u 0 ; 0 ) ≤ d ( ∞ ), where d ( ∞ ) is a nonnegative constant; (iii) 0 < J ( u 0 ; 0 ) ≤ C ρ ( 0 ), where ρ ( 0 ) involves the L 2 -norm or H 0 1 -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.