Blow-up and global existence for a nonlocal degenerate parabolic system ✩

This paper investigates the blow-up and global existence of nonnegative solutions of the system ut = Δum + a v pα , vt = Δvn +b u q β, (x,t)∈Ω ×(0,T ) with homogeneous Dirichlet boundary data, where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω, m,n > 1, α,β 1, p, q, a, b > 0 and · αα ≡ Ω | · |α dx. It is proved that if pq < mn every nonnegative solution is global, whereas if pq > mn, there exist both global and blow-up nonnegative solutions. When pq = mn, we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain is large enough that is, if it contains a sufficiently large ball, there exists no global solution. In particular, when p = n = α, q = m = β, we show that every positive solution exists globally iff Ω ϕ(x)dx 1/√ab, where ϕ(x) is the unique positive solution of the linear elliptic problem −Δϕ(x) = 1, x ∈Ω; ϕ(x) = 0, x ∈ ∂Ω.  2002 Elsevier Science (USA). All rights reserved.