Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term

In this paper, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear source term ut-Δu-Δut+∫0tg(t-τ)Δu(τ)dτ=|u|p-2u,inΩ×(0,T), where Ω is a bounded domain in Rn (n≥1) with a Dirichlet boundary condition. Under suitable assumptions on the initial data u0 and the relaxation function g, we obtain the global existence and finite time blow-up of solutions with initial data at low energy level (i.e. J(u(0))≤d(∞)), by using the Galerkin method, the concavity method and an improved potential well method involving time t. We also derive the upper bounds for the blow-up time. Finally, we obtain the existence of solutions which blow up in finite time with initial data at arbitrary energy level.