Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems with p-Laplacian with Nonlocal Sources. (19th April 2007) - Cui, Zhoujin; Yang, Zuodong
Abstract : 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 ⊂ ℝ 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 ∈ ℝ 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.