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.

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.

scholarly journals Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

2018 ◽  
Vol 148 (5) ◽  
pp. 957-981
Author(s):  
Tomás Caraballo ◽  
Marta Herrera-Cobos ◽  
Pedro Marín-Rubio
Keyword(s):  
Existence And Uniqueness ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Pullback Attractors ◽  
Non Local ◽  
Bounded Sets

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

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