Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms

Abstract

We focus on the global well-posedness of the system of nonlinear wave equations u t t − Δ u + ( d | u | k + e | v | l ) | u t | m − 1 u t = f 1 ( u , v ) v t t − Δ v + ( d ′ | v | θ + e ′ | u | ρ ) | v t | r − 1 v t = f 2 ( u , v ) , in a bounded domain Ω ⊂ R n , n = 1 , 2 , 3 , with Dirichlét boundary conditions. The nonlinearities f 1 ( u , v ) and f 2 ( u , v ) act as a strong source in the system. Under some restriction on the parameters in the system we obtain several results on the existence of local solutions, global solutions, and uniqueness. In addition, we prove that weak solutions to the system blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms.