We show that the reaction-diffusion system ut=Δφ(u)+f(v), vt=Δψ(v)+g(u), with homogeneous Neumann boundary conditions, has a positive global solution on Ω×[0,∞) if and only if ∫∞ds/f(F−1(G(s)))=∞ (or, equivalently, ∫∞ds/g(G−1(F(s)))=∞), where F(s)=∫0sf(r)dr and G(s)=∫0sg(r)dr. The domain Ω⊆ℝN(N≥1) is bounded with smooth boundary. The functions φ, ψ, f, and g are nondecreasing, nonnegative C([0,∞)) functions satisfying φ(s)ψ(s)f(s)g(s)>0 for s>0 and φ(0)=ψ(0)=0. Applied to the special case f(s)=sp and g(s)=sq, p>0, q>0, our result proves that the system has a global solution if and only if pq≤1.
International Journal of Mathematics and Mathematical Sciences
Lair, Alan V., "A Necessary and Sufficient Condition for Global Existence for a Quasilinear Reaction-diffusion System" (2005). Faculty Publications. 248.