Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems with p-Laplacian with Nonlocal Sources. (19th April 2007)

• Record Type: Journal Article
• Authors: Cui, Zhoujin; Yang, Zuodong
• Other Names: Castro Alfonso Academic Editor.
• Is Part Of: International journal of mathematics and mathematical sciences. Voume 2007(2007)
• Publication Date: 2007-04-19
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1155/2007/34301
• Languages: English
• ISSNs: 0161-1712

Abstract

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.