Existence and nonexistence of global solutions of some non-local degenerate parabolic systems

This paper establishes a new criterion for global existence and nonexistence of positive solutions of the non-local degenerate parabolic system ut = v(p) (Deltau+a integral(Omega) vdx), vt = u(q) (Deltav+b integral(Omega) udx), x is an element of Omega, t > 0, with homogeneous Dirichlet boundary conditions, where Omega subset of R-N is a bounded domain with a smooth boundary partial derivativeOmega and p, q, a, b are positive constants. For all initial data, it is proved that there exists a global positive solution iff integral(Omega)phi(x)dx less than or equal to 1/rootab, where phi(x) is the unique positive solution of the linear elliptic problem -Deltaphi(x)=1, x is an element of Omega; phi(x)=0, x is an element of partial derivativeOmega.