scholarly journals A Class of Nonlocal Coupled Semilinear Parabolic System with Nonlocal Boundaries

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Hong Liu ◽  
Haihua Lu
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Parabolic System ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Dirichlet Boundary Conditions ◽  
Nonlocal Sources ◽  
Semilinear Parabolic System ◽  
Semilinear Parabolic ◽  
Existence Criteria

We investigate the positive solutions of the semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions. The blow-up and global existence criteria are obtained.

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

2007 ◽  
Vol 2007 ◽  
pp. 1-17 ◽  
Author(s):  
Zhoujin Cui ◽  
Zuodong Yang
Keyword(s):  
Global Existence ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Elliptic Systems ◽  
Dirichlet Boundary Conditions ◽  
Local Theory ◽  
Smooth Bounded Domain ◽  
Nonlocal Sources ◽  
Nonexistence Result ◽  
Evolution Systems

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⊂ℝN, 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∈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.

scholarly journals Blow-Up and Global Existence for a Degenerate Parabolic System with Nonlocal Sources

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Ling Zhengqiu ◽  
Wang Zejia
Keyword(s):  
Global Existence ◽  
Critical Exponent ◽  
Parabolic System ◽  
Blow Up ◽  
Nonnegative Solution ◽  
Degenerate Parabolic System ◽  
Nonlocal Sources ◽  
Nonnegative Solutions ◽  
Large Ball ◽  
Degenerate Parabolic

This paper investigates the blow-up and global existence of nonnegative solutions for a class of nonlocal degenerate parabolic system. By using the super- and subsolution techniques, the critical exponent of the system is determined. That is, ifPc=p1q1−(m−p2)(n−q2)<0, then every nonnegative solution is global, whereas ifPc>0, there are solutions that blowup and others that are global according to the size of initial valuesu0(x)andv0(x). WhenPc=0, we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain large enough that is, if it contains a sufficiently large ball, there is no global solution.

Über die C2-Kompaktheit der Bahn von Lösungen semflinearer parabolischer Systeme

1982 ◽  
Vol 93 (1-2) ◽  
pp. 99-103 ◽  
Author(s):  
Reinhard Redlinger
Keyword(s):  
Boundary Conditions ◽  
Regular Solution ◽  
Parabolic System ◽  
Lipschitz Continuous ◽  
Semilinear Parabolic System ◽  
Locally Lipschitz ◽  
Semilinear Parabolic ◽  
Locally Lipschitz Continuous

SynopsisThe semilinear parabolic systemut+A(x, D)u=g(u) in (0, ∞) × Ω, Ω⊂ℝnbounded,u∈ ℝN, with homogeneous boundary conditionsB(x, D)u=0 on (0, ∞)×∂Ω is considered. The non-linearitygis assumed to be locally Lipschitz-continuous. It is shown that the orbit of a bounded regular solutionuis relatively compact in.

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