Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems with p -Laplacian with Nonlocal Sources

This paper deals with p -Laplacian systems u t − div ( | ∇ u | p − 2 ∇ u ) = ∫ Ω v α ( x , t ) d x , x ∈ Ω , t > 0 , v t − div ( | ∇ v | q − 2 ∇ v ) = ∫ Ω u β ( x , t ) d x , x ∈ Ω ,  t > 0, with null Dirichlet boundary conditions in a smooth bounded domain Ω ⊂ ℠N , 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 = B R = { x ∈ ℠N : | x | < R }  ( R > 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.