Global existence and blow-up solutions for doubly degenerate parabolic system with nonlocal source

This paper deals with the following nonlocal doubly degenerate parabolic systemut−div(|∇um|p−2∇um)=a∫Ωuα1(x,t)vβ1(x,t)dx,vt−div(|∇vn|q−2∇vn)=b∫Ωuα2(x,t)vβ2(x,t)dx with null Dirichlet boundary conditions in a smooth bounded domain Ω⊂RNΩ⊂RN, where m,n⩾1m,n⩾1, p,q>2p,q>2, αi,βi⩾0αi,βi⩾0, i=1,2i=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 non-Newton polytropic filtration systems with the homogeneous Dirichlet boundary value conditions are obtained. Then under appropriate hypotheses, we establish local theory of the solutions and prove that the solution either exists globally or blows up in finite time. In the special case, β1=n(q−1)−β2β1=n(q−1)−β2, α2=m(p−1)−α1α2=m(p−1)−α1, we also give a criterion for the solution to exist globally or blow up in finite time, which depends on a, b   and ζ(x)ζ(x), ϑ(x)ϑ(x) as defined in the main results.