CRITICAL BLOWUP AND GLOBAL EXISTENCE NUMBERS FOR A WEAKLY COUPLED SYSTEM OF REACTION-DIFFUSION EQUATIONS

Let D subset of R(N) be either all of R(n) or else a cone in R(N) whose vertex we may take to be at the origin, without loss of generality. Let p(i), q(j), i = 1, 2, be nonnegative with 0 < p(1) + q(1) less than or equal to p(2) + q(2). We consider the long-time behavior of nonnegative solutions of the system (S) u(t) = Delta u + u(p1)v(q1), v(t) = Delta v + u(p2),v(q2) in D x [0, infinity] with u(0) = v(0) = 0 on partial derivative D, (u, v)(t)(x, 0) = (v(0), u(0))(t)(x), u(0), v(0) greater than or equal to 0, u(0), v(0) is an element of L(infinity) (D). We obtain Fujita-type global existence-global nonexistence theorems for (S) analogous to the classical result of FUJITA and others for the initial-value problem for u(t) = Delta u + u(p), u(x, 0) = u(0)(x) greater than or equal to 0. The principal result in the case D = R(N) and p(2)q(1) > 0 is that when p(1) greater than or equal to 1, the system behaves like the single equation u(t) = Delta u + u(p1+q1) with respect to Fujita-type blowup theorems, whereas if p(1) < 1, the behavior of the system is more complicated. Some of the results extend those of ESCOBEDO & HERRERO when D = R(N) and of LEVINE when D is a cone. These authors considered (S) in the case of p(1) = q(2) = 0. An example of nonuniqueness is also given.