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This paper deals with p-Laplacian systems ut−div(|∇u|p−2∇u)=∫Ωvα(x, t)dx, x∈Ω, t>0, vt−div(|∇v|q−2∇v)=∫Ωuβ(x,t)dx, x∈Ω, t>0, with null Dirichlet boundary conditions in a smooth bounded domain Ω⊂ℝN, where p,q≥2, α,β≥1. We first get the nonexistence result for related elliptic systems of nonincreasing positive solutions. Secondly by using this nonexistence result, blow up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained under Ω=BR={x∈ℝN:|x|0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time.

This paper investigates the local existence of the nonnegative solution and the finite time blow-up of solutions and boundary layer profiles of diffusion equations with nonlocal reaction sources; we also study the global existence and that the rate of blow-up is uniform in all compact subsets of the domain, the blow-up rate of |u(t)|∞ is precisely determined.