Existence and nonexistence of the global solution for the semilinear parabolic system on Riemannian manifold

In this paper, we study the global existence and nonexistence of solutions to the following semilinear parabolic system {u(t) - Delta u = v(p) in M-n x (0, infinity), v(t) - Delta v = u(p) in M-n x (0, infinity), u(x, 0) = u(0)(x) in M-n, v(x, 0) = v(0)(x) in M-n, where M-n (n >= 3) is a non-compact complete Riemannian manifold, Delta is the Laplace-Beltrami operator. We assume that both u(0)(x) and v(0)(x) are nonnegative, smooth and bounded functions, constants p, q >= 1. (C) 2009 Elsevier Ltd. All rights reserved.