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

摘要

This paper deals withp-Laplacian systemsut?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 domainΩ??N, wherep,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 abovep-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained underΩ=BR={x∈?N:|x|0). Then under appropriate hypotheses, we establish local theory of thesolutions and obtain that the solutions either exist globally or blow up in finite time.