Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping

This paper concerns with the Cauchy problem for the damped nonlinear hyperbolic system {utt−△u+ut=|v|p,(t,x)∈R+1×RN,vtt−△v+vt=|u|q,(t,x)∈R+1×RN,u(0,x)=u0(x),ut(0,x)=u1(x),x∈RN,v(0,x)=v0(x),vt(0,x)=v1(x),x∈RN, where p,q≥1p,q≥1 and satisfy pq>1pq>1. We shall prove that if max{1+ppq−1,1+qpq−1}