A Fujita‐type global existence—global non‐existence theorem for a system of reaction diffusion equations with differing diffusivities

Abstract In this paper we condiser non‐negative solutions of the initial value problem in ℝ N for the system\documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm u}_{{\rm t = }} {\rm \delta \Delta u + v}^{\rm p}, $$\end{document}\documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm v}_{{\rm t = }} {\rm \Delta v + u}^{\rm q}, $$\end{document}where 0 ⩽ δ ⩽ 1 and pq > 0. We prove the following conditions. Suppose min( p , q )≥1 but pq 1.(a) If δ = 0 then u = v =0 is the only non‐negative global solution of the system. (b) If δ>0, non‐negative non‐globle solutions always exist for suitable initial values. (c) If 0<⩽1 and max(α, β) ≥ N /2, where q α = β + 1, p β = α + 1, then the conclusion of (a) holds. (d) If N > 2, 0 < δ ⩽ 1 and max (α β) < ( N ‐ 2)/2, then global, non‐trivial non‐negative solutions exist which belong to L ∞ (ℝ N ×[0, ∞]) and satisfy 0 < u (X, t ) ⩽ c ∣x∣ −2α and 0 < v (X, t ) ⩽ c ∣x∣ −2bT for large ∣x∣ for all t > 0, where c depends only upon the initial data. (e) Suppose 0 > δ 1 and max (α, β) < N /2. If N > = 1,2 or N > 2 and max ( p , q )⩽ N /( N ‐2), then global, non‐trivial solutions exist which, after makinng the standard ‘hot spot’ change of variables, belong to the weighted Hilbert space H 1 ( K ) where K (x) exp(¼∣x∣ 2 ). They decay like e [max(α,β)‐( N /2)+ε] t for every ε > 0. These solutions are classical solutions for t > 0. (f) If max (α, β) < N /2, then threre are global non‐tivial solutions which satisfy, in the hot spot variables\documentclass{article}\pagestyle{empty}\begin{document}$$ \max (u,v)(x,t) \le c(u_0,v_0){\rm e}^{ - \frac{1}{4}|x|^2 } {\rm e}^{[\max (\alpha, \beta) - N/2) + \varepsilon]t}, $$\end{document}where where 0 < ε = ε( u 0 , v 0 ) < ( N /2)−;max(α, β). Suppose min( p , q ) ⩽ 1. (g) If pq ≥ 1, all non‐negative solutions are global. Suppose min( p , q ) < 1. (h) If pg > 1 and δ = 0, than all non‐trivial non‐negative maximal solutions are non‐global. (i) If 0 < δ ⩽ 1, pq > 1 and max(α,β)≥ N /2 all non‐trivial non‐negative maximal solutions are non‐global. (j) If 0 < δ ≥ 1, pq > 1 and max(α,β) < N /2, there are both global and non‐negative solutions.We also indicate some extensions of these results to moe general systems and to othere geometries.