Global existence and blow-up solutions and blow-up estimates for a non-local quasilinear degenerate parabolic system

This paper deals with p-Laplacian systemsut-div(|∇u|p-2∇u)=a∫Ωuα1(x,t)vβ1(x,t)dx,x∈Ω,t>0,vt-div(|∇v|q-2∇v)=b∫Ωuα2(x,t)vβ2(x,t)dx,x∈Ω,t>0with null Dirichlet boundary conditions in a smooth bounded domain Ω⊂RNΩ⊂RN, where p,q>1p,q>1, αi,βi⩾0,i=1,2, and a,b>0a,b>0 are positive constants. We first get the non-existence result for a related elliptic systems of non-increasing positive solutions. Secondly by using this non-existence 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)Ω=BR={x∈RN:|x|0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exists globally or blow-up in finite time.