2011
DOI: 10.1016/j.jmaa.2011.03.053
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Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems

Abstract: A family of compact and positively invariant sets with uniformly bounded fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the process is constructed. The existence of such a family, called a pullback exponential attractor, is proved for a nonautonomous semilinear abstract parabolic Cauchy problem. Specific examples will be presented in the forthcoming Part II of this work.

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Cited by 47 publications
(62 citation statements)
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“…We refer to [3,4,5,9,10,14,22,24,26,49] for the concepts and some of the existing results in the theory of nonautonomous dynamical systems, especially on the topics of pullback attractors and pullback exponential attractors. Recall that these concepts are rooted in the theory of global attractors and other invariant attracting sets for the autonomous infinite-dimensional dynamical systems [7,28,32,35,40,44,45,46] and the theory of exponential attractors or sometimes called inertial sets [12,13,25,28,43].…”
Section: Pullback Attractor and Pullback Exponential Attractormentioning
confidence: 99%
“…We refer to [3,4,5,9,10,14,22,24,26,49] for the concepts and some of the existing results in the theory of nonautonomous dynamical systems, especially on the topics of pullback attractors and pullback exponential attractors. Recall that these concepts are rooted in the theory of global attractors and other invariant attracting sets for the autonomous infinite-dimensional dynamical systems [7,28,32,35,40,44,45,46] and the theory of exponential attractors or sometimes called inertial sets [12,13,25,28,43].…”
Section: Pullback Attractor and Pullback Exponential Attractormentioning
confidence: 99%
“…In Part I of this work (see [5]) we have constructed a pullback exponential attractor for an evolution process. By this we mean a family of compact and positively invariant sets with uniformly bounded fractal dimension which under the evolution process at a uniform exponential rate pullback attract bounded subsets of the phase space.…”
Section: Pullback Exponential and Global Attractors For Semilinear Pamentioning
confidence: 99%
“…Moreover, we have formulated conditions under which the mentioned abstract results apply to nonautonomous semilinear parabolic problems. For completeness we recall here the main result (see [5,Theorem 3.6]) and refer the reader for the proof and details to Part I of this work.…”
Section: Pullback Exponential and Global Attractors For Semilinear Pamentioning
confidence: 99%
“…The method used to prove the existence of the nonautonomous exponential attractor is similar to that in [Efendiev et al, 2005;Málek & Pražák, 2002]. The following points are notable in this paper: (1) the methods presented in [Efendiev et al, 2005;Langa et al, 2010] are not utilized to prove the existence of the pullback exponential attractor for the nonautonomous reactiondiffusion equations in strong space L p (Ω)(p > 2) because of two reasons: (a) space L p (Ω)(p > 2) is the most regular space, hence compact embedding does not hold; and (b) the Lipschitz continuous condition does not exist between spaces L 2 (Ω) and L p (Ω)(p > 2); and (2) all the results obtained in [Eden et al, 1994;Efendiev et al, 2005;Langa et al, 2010;Czaja & Efendiev, 2011] need the Lipschitz continuous condition for time to prove the existence of the nonautonomous exponential attractor.…”
Section: Introductionmentioning
confidence: 99%