Blow-up for coupled nonlinear wave equations with damping and source

Abstract

We consider the system of nonlinear wave equations { u t t + u t + | u t | m − 1 u t = div ( ρ 1 ( | ∇ u | 2 ) ∇ u ) + f 1 ( u , v ) , ( x , t ) ∈ Ω × ( 0 , T ) , v t t + v t + | v t | r − 1 v t = div ( ρ 2 ( | ∇ v | 2 ) ∇ v ) + f 2 ( u , v ) , ( x , t ) ∈ Ω × ( 0 , T ) , with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions f 1 , f 2 , ρ 1 , ρ 2 , parameters r , m and the initial data, the result on blow-up of solutions and upper bound of blow-up time are given.