A FUJITA-TYPE GLOBAL EXISTENCE - GLOBAL NONEXISTENCE THEOREM FOR A SYSTEM OF REACTION-DIFFUSION EQUATIONS WITH DIFFERING DIFFUSIVITIES

In this paper we consider non-negative solutions of the initial value problem in R(N) for the system u(t) = delta DELTAu + upsilon(p), upsil on(t) = DELTAupsilon + u(q), where 0 less-than-or-equal-to delta less- than-or-equal-to 1 and pq > 0. We prove the following conditions. Supp ose min(p, q) greater-than-or-equal-to 1 but pq > 1. (a) If delta = 0 then u = upsilon = 0 is the only non-negative global solution of the s ystem. (b) If delta > 0, non-negative non-global solutions always exis t for suitable initial values. (c) If 0 < delta less-than-or-equal-to 1 and max(alpha, beta) greater-than-or-equal-to N/2, where qalpha = be ta + 1, pbeta = alpha + 1, then the conclusion of (a) holds. (d) If N > 2, 0 < delta less-than-or-equal-to 1 and max(alpha, beta) < (N - 2)/ 2, then global, non-trivial non-negative solutions exist which belong to L(infinity) (R(N) x [0, infinity)) and satisfy 0 < u(x,t) less-than -or-equal-to c Absolute value of x -2alpha and 0 < upsilon(x, t) less- than-or-equal-to c Absolute value of x -2beta for large Absolute value of x for all t > 0, where c depends only upon the initial data. (e) S uppose 0 < delta less-than-or-equal-to 1 and max(alpha,beta) < N/2. If N = 1, 2 or N > 2 and max(p,q) less-than-or-equal-to N/(N - 2), then global, non-trivial solutions exist which, after making the standard ' hot spot' change of variables, belong to the weighted Hilbert space H- 1(K) where K(x) = exp(1/4 Absolute value of x 2). They decay like e[ma x(alpha, beta) -(N/2)+epsilon]t for every epsilon > 0. These solutions are classical solutions for t > 0. (f) If max(alpha, beta) < N/2, the n there are global non-trivial solutions which satisfy, in the hot spo t variables, max(u, v)(x, t) less-than-or-equal-to c(u0, upsilon0)e-1/ 4 Absolute value of 2 e(max(alpha,beta)-(N/2)+epsilon)t, where 0 < eps ilon = epsilon(u0, upsilon0) < (N/2) - max(alpha, beta). Suppose min ( p,q) less-than-or-equal-to 1. (g) If pq less-than-or-equal-to 1, all n on-negative solutions are global. Suppose min(p, q) < 1. (h) If pq > 1 and delta = 0, then all non-trivial non-negative maximal solutions ar e non-global. (i) If 0 < delta less-than-or-equal-to 1, pq > 1 and max (alpha, beta) greater-than-or-equal-to N/2 all non-trivial non-negativ e maximal solutions are non-global. (j) If 0 < delta less-than-or-equa l-to 1, pq > 1 and max(alpha, beta) < N/2, there are both global and n on-global non-negative solutions. We also indicate some extensions of these results to more general systems and to other geometries.