Nonexistence of global solutions for a nonlocal nonlinear hyperbolic system with linear damping

This article concerns the Cauchy problem for the damped nonlinear hyperbolic system Ïμutt+(-Δ)αu+ut=vp,t>0,x∈RN, u>0,v>0,Ïμvtt+(-Δ)αv+vt=uq,t>0,x∈RN,u>0, v>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈RN,v(x,0)=v0(x),vt(x,0)=v1(x), x∈RN, where Ïμ > 0 is a small parameter, 0 < α ≤ 1,0 < β ≤ 1,p,q ≥ 1 satisfying pq > 1, and N ≥ 1 is an integer.It is proved that if N/2α < max((p + 1)/(pq - 1),(q + 1)/(pq - 1)), then every weak solution does not exist globally whenever the initial data satisfy ∫RN{u0(x)+u1(x)}dx>0 or ∫RN(v0(x)+v1(x))dx>0.