Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations

We examine the parabolic system of three equations ${\displaystyle u_t}$ - Δu = ${\displaystyle v^p}$, ${\displaystyle v_t}$ - Δv = ${\displaystyle w^q}$, ${\displaystyle w_t}$ - Δw = ${\displaystyle u^r}$, x ∈ ${\displaystyle {ℝ}^N}$, t > 0 with p, q, r positive numbers, N ≥ 1, and nonnegative, bounded continuous initial values. We obtain global existence and blow up unconditionally (that is, for any initial data). We prove that if pqr ≤ 1 then any solution is global; when pqr > 1 and max(α,β,γ) ≥ N/2 (α, β, γ are defined in terms of p, q, r) then every nontrivial solution exhibits a finite blow up time.