In this paper we investigate blow up property of solutions for a system of nonlinear wave equations with nonlinear dissipations and positive initial energy in a bounded domain in R3. Our result improves and extends earlier results in the literature such as the ones in [Zhou, J. and Mu, C., The lifespan for 3D quasilinear wave equations with nonlinear damping terms, Nonlinear Anal., 74 (2011), 5455–5466] and [Pis¸kin, E., Uniform decay and blow-up of solutions for coupled nonlinear Klein-Gordon equations with nonlinear damping terms, Math. Meth. Appl. Scie., 37 (2014), No. 18, 3036–3047] in which the nonexistence results obtained only for negative initial energy or the one in [Ye, Y., Global existence and nonexistence of solutions for coupled nonlinear wave equations with damping and source terms, Bull. Korean Math. Soc., 51 (2014), No. 6, 1697–1710] where blow up results have been not addressed. Estimate for the lower bound of the blow up time is also given.
A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy