scholarly journals Regularity of Global Attractor for the Reaction-Diffusion Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Hong Luo
Keyword(s):  
Sobolev Space ◽  
Diffusion Equation ◽  
Existence Theorem ◽  
Global Attractor ◽  
Reaction Diffusion ◽  
Iteration Procedure ◽  
Reaction Diffusion Equation ◽  
Bounded Subset ◽  
Regularity Estimates ◽  
Regularity Of Global Attractor

By using an iteration procedure, regularity estimates for the linear semigroups, and a classical existence theorem of global attractor, we prove that the reaction-diffusion equation possesses a global attractor in Sobolev spaceHkfor allk>0, which attracts any bounded subset ofHk(Ω) in theHk-norm.

Global Attractor of the Liquid Helium-4 System in Hk Space

2013 ◽  
Vol 444-445 ◽  
pp. 731-737
Author(s):  
Zhi Bo Hou ◽  
Li Mei Li
Keyword(s):  
Existence Theorem ◽  
Liquid Helium ◽  
Global Attractor ◽  
Iteration Procedure ◽  
Helium 4 ◽  
Bounded Set ◽  
Regularity Estimates

In this paper, by using an iteration procedure, regularity estimates of the linear semi-groups and a generalized existence theorem of global attractor, we prove that the liquid helium-4 system possesses a global attractor in space for all , which attracts any bounded set of in the-norm.

scholarly journals DYNAMICS OF A REACTION–DIFFUSION EQUATION WITH A DISCONTINUOUS NONLINEARITY

2006 ◽  
Vol 16 (10) ◽  
pp. 2965-2984 ◽  
Author(s):  
JOSÉ M. ARRIETA ◽  
ANÍBAL RODRÍGUEZ-BERNAL ◽  
JOSÉ VALERO
Keyword(s):  
Nonlinear Dynamics ◽  
Diffusion Equation ◽  
Fixed Points ◽  
Global Attractor ◽  
Nonlinear Term ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Discontinuous Nonlinearity ◽  
Smooth Approximations

We study the nonlinear dynamics of a reaction–diffusion equation where the nonlinearity presents a discontinuity. We prove the upper semicontinuity of solutions and the global attractor with respect to smooth approximations of the nonlinear term. We also give a complete description of the set of fixed points and study their stability. Finally, we analyze the existence of heteroclinic connections between the fixed points, obtaining information on the fine structure of the global attractor.

scholarly journals Global attractor for a 3D reaction-diffusion equation

2014 ◽  
Vol 9 ◽  
pp. 13-18
Author(s):  
Xiaosong Wang ◽  
Hongjun Wang ◽  
Lingrui Zhang
Keyword(s):  
Diffusion Equation ◽  
Global Attractor ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation

scholarly journals A global solution for a reaction-diffusion equation on bounded domains

2018 ◽  
Vol 3 (1) ◽  
pp. 15-22 ◽  
Author(s):  
Farhad Khellat ◽  
Mahmud Beyk Khormizi
Keyword(s):  
Diffusion Equation ◽  
Global Solution ◽  
Global Attractor ◽  
Bounded Domain ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Bounded Domains ◽  
Uniform Bound ◽  
Bounded Perturbation

AbstractIn the literature, it has been proved the existence of a pullback global attractor for reaction-diffusion equation on a bounded domain and under some conditions, a uniform bound on the dimension of its sections. Using those results and putting a bound on the diameter of the domain, we proved that the pullback global attractor consists only of one global solution. As an application to this result, a bounded perturbation of Chafee-Infante equation has been studied.

scholarly journals Global dynamics of solutions for a sixth-order parabolic equation describing continuum evolution of film-free surface

2022 ◽  
Vol 27 (1) ◽  
pp. 19-37
Author(s):  
Ning Duan ◽  
Xiaopeng Zhao
Keyword(s):  
Free Surface ◽  
Parabolic Equation ◽  
Diffusion Equation ◽  
Existence Theorem ◽  
Global Attractor ◽  
Global Attractors ◽  
Sixth Order ◽  
Global Dynamics ◽  
Regularity Estimates ◽  
Iteration Technique

This paper is concerned with a sixth-order diffusion equation, which describes continuum evolution of film-free surface. By using the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors we verified the existence of global attractor for this surface diffusion equation in the spaces H3(Ω) and fractional-order spaces Hk(Ω), where 0 ≤ k < ∞.

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