International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 34301, 17 pages
Research Article

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

Zhoujin Cui1 and Zuodong Yang1,2

1Institute of Mathematics, School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 210097, China
2College of Zhongbei, Nanjing Normal University, Nanjing 210046, China

Received 20 September 2006; Accepted 21 February 2007

Academic Editor: Alfonso Castro

Copyright © 2007 Zhoujin Cui and Zuodong Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper deals with p-Laplacian systems utdiv(|u|p2u)=Ωvα(x, t)dx, xΩ, t>0, vtdiv(|v|q2v)=Ωuβ(x,t)dx, xΩ, t>0, with null Dirichlet boundary conditions in a smooth bounded domain ΩN, where p,q2, α,β1. 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=BR={xN:|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.