This work is concerned with a system of nonlinear wave equations with nonlinear damping and source terms acting in both equations. We will prove that the solution of our considered system blows up in finite time provided that the initial data are large enough. This result extends a previous result in  to a large class of initial data. The key ingredient in the proof is a method used in  with necessary modifications imposed by the nature of our problem.
Belkacem Said-Houari. "Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms." Differential Integral Equations 23 (1/2) 79 - 92, January/February 2010. https://doi.org/10.57262/die/1356019388