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

• Main Authors: Zhoujin Cui, Zuodong Yang
• Format: Article
• Language: English
• Published: Hindawi Limited 2007-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Online Access: http://dx.doi.org/10.1155/2007/34301

This paper deals with p-Laplacian systems ut−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 domain Ω⊂ℝN, 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=BR={x∈ℝN:|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.