Global existence and finite time blow up for a degenerate reaction-diffusion system

This paper investigates the blow-up and global existence of solutions of the degenerate reaction-diffusion systemut=Δum+uαvp,vt=Δvn+uqvβ,(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⩾0 and p,q>0. It is proved that if m>α,n>β and pq<(m-α)(n-β) every nonnegative solution is global, whereas if m<α or n<β or pq>(m-α)(n-β), there exist both global and blow up nonnegative solutions. When m>α, n>β and pq=(m-α)(n-β), we show that there exists λ*⩽1 which depends on the parameters p,q,m,n,α,β such that all positive solutions are global if λ1>λ*, while if λ1<1/λ* all positive solutions blow up in finite time, where λ1 is the first Dirichlet eigenvalue for the Laplacian on Ω.