An exponential growth condition in for the pullback attractor of a non-autonomous reaction–diffusion equation

2010 ◽  
Vol 72 (11) ◽  
pp. 4071-4075 ◽  
Author(s):  
M. Anguiano ◽  
T. Caraballo ◽  
J. Real
Keyword(s):  
Diffusion Equation ◽  
Growth Condition ◽  
Exponential Growth ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Pullback Attractor

scholarly journals The Existence of WeakD-Pullback Exponential Attractor for Nonautonomous Dynamical System

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Yongjun Li ◽  
Xiaona Wei ◽  
Yanhong Zhang
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
External Force ◽  
Exponential Growth ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Nonautonomous Dynamical System

First, for a processU(t,τ)∣t≥τ, we introduce a new concept, called the weakD-pullback exponential attractor, which is a family of setsM(t)∣t≤T, for anyT∈R, satisfying the following: (i)M(t)is compact, (ii)M(t)is positively invariant, that is,U(t,τ)M(τ)⊂M(t), and (iii) there existk,l>0such thatdist(U(t,τ)B(τ),M(t))≤ke-(t-τ); that is,M(t)pullback exponential attractsB(τ). Then we give a method to obtain the existence of weakD-pullback exponential attractors for a process. As an application, we obtain the existence of weakD-pullback exponential attractor for reaction diffusion equation inH01with exponential growth of the external force.

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.

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