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

Abstract

AbstractIn this paper we condiser non‐negative solutions of the initial value problem in ℝN for the system where 0 ⩽ δ ⩽ 1 and pq > 0. We prove the following conditions.Suppose min(p,q)≥1 but pq1. If δ = 0 then u=v=0 is the only non‐negative global solution of the system. If δ>0, non‐negative non‐globle solutions always exist for suitable initial values. If 0<⩽1 and max(α, β) ≥ N/2, where qα = β + 1, pβ = α + 1, then the conclusion of (a) holds. 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∞(ℝN×[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. 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 H1 (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. If max (α, β) < N/2, then threre are global non‐tivial solutions which satisfy, in the hot spot variables where where 0 < ε = ε(u0, v0) < (N/2)−;max(α, β).Suppose min(p, q) ⩽ 1. If pq ≥ 1, all non‐negative solutions are global.Suppose min(p, q) < 1. If pg > 1 and δ = 0, than all non‐trivial non‐negative maximal solutions are non‐global. If 0 < δ ⩽ 1, pq > 1 and max(α,β)≥ N/2 all non‐trivial non‐negative maximal solutions are non‐global. 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.