Semilinear reaction-diffusion systems with nonlocal sources

This paper investigates the homogeneous Dirichlet boundary value problem uit − δuit = ∏j=1n∫ω uJPij dx, i = 1, 2, …, n in a bounded domain Ω ⊂ RN, where pij ≥ 0 (1 ≤ i, j ≤ n) are constants. Denote by I the identity matrix and P = (pij), which is assumed to be irreducible. It is shown that if I - P is an M-matrix, every nonnegative solution is global, whereas if I - P is not an M-matrix, there exist both global and nonglobal nonnegative solutions.