Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

Abstract

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations u t t − Δ u + ∫ 0 t g 1 ( t − τ ) Δ u ( τ ) d τ + | u t | m − 1 u t = f 1 ( u , v ) , v t t − Δ v + ∫ 0 t g 2 ( t − τ ) Δ v ( τ ) d τ + | v t | r − 1 v t = f 2 ( u , v ) in Ω × ( 0 , T ) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in R n , n = 1 , 2 , 3 . Under suitable assumptions on the functions g i ( ⋅ ) , f i ( ⋅ , ⋅ ) ( i = 1 , 2 ) , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.