ON EXISTENCE, UNIFORM DECAY RATES AND BLOW UP FOR SOLUTIONS OF SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH DAMPING AND SOURCE TERMS

This paper is concerned with the study of the nonlinearly damped system of wave equations with Dirichlet boundary conditions: u(tt )- Delta u + vertical bar u(t)vertical bar(m-1)u(t) = F-u(u, v) in Omega x (0, infinity), v(tt )- Delta v + vertical bar v(t)vertical bar(r-1)v(t) = F-v(u, v) in Omega x (0, infinity), where Omega is a bounded domain in R-n, n = 1,2,3 with a smooth boundary partial derivative Omega = Gamma and F is a C-1 function given by F(u, v) = alpha vertical bar u + v vertical bar(p+1) + 2 beta vertical bar uv vertical bar(p+1/2). Under some conditions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain several results on the global existence, uniform decay rates, and blow up of solutions in finite time when the initial energy is nonnegative.