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

In this paper we condiser non-negative solutions of the initial value problem in ?^N for the system where 0 ? δ ? 1 and pq > 0. We prove the following conditions. Suppose min(p,q)≥1 but pq1.

• (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 qα = β + 1, pβ = 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^∞(?^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 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.
• (f) 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.
• (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.