Global existence and blow-up for the degenerate and singular nonlinear parabolic system with a nonlocal source

The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained for the degenerate and singular parabolic system ut−(xαux)x=∫0aum(x,t)vn(x,t)dx,vt−(xβvx)x=∫0avp(x,t)uq(x,t)dx,in (0,a)×(0,T), where T≤∞,a>0T≤∞,a>0 are constants and m,n,p,qm,n,p,q are positive real numbers. Furthermore, under certain conditions it is proved that the blow-up set of the solution is the entire interval [0,a][0,a]. And we also obtain the blow-up rate under the condition α=βα=β.