scholarly journals EXISTENCE OF PULLBACK ATTRACTOR FOR A REACTION–DIFFUSION EQUATION IN SOME UNBOUNDED DOMAINS WITH NON-AUTONOMOUS FORCING TERM IN H-1

2010 ◽  
Vol 20 (09) ◽  
pp. 2645-2656 ◽  
Author(s):  
MARÍA ANGUIANO ◽  
TOMÁS CARABALLO ◽  
JOSÉ REAL
Keyword(s):  
Diffusion Equation ◽  
Poincaré Inequality ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Pullback Attractor ◽  
Poincare Inequality ◽  
Main Concept ◽  
Asymptotic Compactness ◽  
Forcing Term

The existence of a pullback attractor in L2(Ω) for the following non-autonomous reaction–diffusion equation [Formula: see text] is proved in this paper, when the domain Ω is not necessarily bounded but satisfying the Poincaré inequality, and [Formula: see text]. The main concept used in the proof is the asymptotic compactness of the process generated by the problem.

Pullback Attractors for a Reaction–Diffusion Equation in a General Nonempty Open Subset of ℝN with Nonautonomous Forcing Term in H−1

2015 ◽  
Vol 25 (12) ◽  
pp. 1550164
Author(s):  
María Anguiano
Keyword(s):  
Diffusion Equation ◽  
Growth Condition ◽  
Open Subset ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonempty Open Subset ◽  
Pullback Attractors ◽  
Main Concept ◽  
Asymptotic Compactness ◽  
Bounded Sets

The existence of minimal pullback attractors in [Formula: see text] for a nonautonomous reaction–diffusion equation, in the frameworks of universes of fixed bounded sets and that given by a tempered growth condition, is proved in this paper, when the domain [Formula: see text] is a general nonempty open subset of [Formula: see text], and [Formula: see text]. The main concept used in the proof is the asymptotic compactness of the process generated by the problem. The relation among these families is also discussed.

Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

2019 ◽  
Vol 150 (2) ◽  
pp. 721-739
Author(s):  
Sergei Trofimchuk ◽  
Vitaly Volpert
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlocal Nonlinearity ◽  
Estimates Of Solutions ◽  
Priori Estimates

AbstractReaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

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