A FUJITA TYPE GLOBAL EXISTENCE-GLOBAL NONEXISTENCE THEOREM FOR A WEAKLY COUPLED SYSTEM OF REACTION-DIFFUSION EQUATIONS

Let D subset-of R(N) be a region with smooth boundary delta-D. Let p . q > 1, p, q greater-than-or-equal-to 1. We consider the system: u(t) = DELTA-u + v(p), v(t) = DELTA-v + u(q) in D x [0, infinity) with u = v = 0 in delta-D x [0, infinity) and u0, v0 nonnegative. Let gamma = max(p, q). We show that if D is R(N), a cone or the exterior of a bounded domain, then there is a number pc(D) such that (a) if (gamma + 1)/(pq - 1) > pc(D) no nontrivial global positive solutions of the system exist while (b) if (gamma + 1)/(pq - 1) < pc(D) both nontrivial global and nonglobal solutions exist. In case D is a cone or D = R(N), (a) holds with equality. An explicit formula for pc(D) is given.