Blow-up of solution for an integro-differential equation with arbitrary positive initial energy

摘要

In this paper, we consider the integro-differential equation u t t − M ( ∥ ∇ u ∥ 2 2 ) Δ u + ∫ 0 t g ( t − τ ) Δ u ( τ ) d τ + u t = f ( u ) $u_{tt}-M(|abla u|^{2}_{2})Delta u+ int_{0}^{t} g(t-au)Delta u(au), dau+ u_{t}=f(u)$ , ( x , t ) ∈ Ω × ( 0 , T ) $(x,t)inOmegaimes(0,T)$ , with initial and Dirichlet boundary conditions. Under suitable assumptions on the functions g and the initial data, a blow-up result with arbitrary positive initial energy is established.