Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications

Carvalho, Alexandre N.; Sonner, Stefanie

doi:10.3934/cpaa.2014.13.1141

Cited by 29 publications

(26 citation statements)

References 17 publications

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“…In what follows, As an application to Theorem 3, we examine the asymptotic behavior of a new version of non-autonomous Chafee-Infante equation respecting the new assumption (12). Some other versions of this equation in stronger spaces have arisen in [5], [3], [2] and [7]. In [3], the unperturbed version of the following equation has been studied and the trivial asymptotic dynamics has been shown when λ < 1.…”

Section: Lemmamentioning

confidence: 99%

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

Khellat

Khormizi

2018

Applied Mathematics and Nonlinear Sciences

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In 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.

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Section: Lemmamentioning

confidence: 99%

“…Consider the problem (1) on a bounded, smooth and convex domain Ω with the assumptions (2)-(5) and(12) and let {A (t) : t ∈ R} be the pullback global attractor in L 2 (Ω) associated to it. If diam(Ω) < π(c 5 + α2 ) −1 , then A (t) = {u(t)} where u is a global solution of the problem.…”

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

Khellat

Khormizi

2018

Applied Mathematics and Nonlinear Sciences

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“…We notice that the evolution mode of states in a random system is, in some sense, similar to the deterministic non-autonomous one and there were several construction methods to obtain a pullback exponential attractor for a (deterministic) process, see [14,15,16,17,20,35,61]. We also notice that a trajectory of a random system is often unbounded in time along the path of sample point with probability 1 which is different from deterministic one [14,15,16,17,20,35,61]. Thus, in general, a simple straightforward extension from deterministic system to random system does not work.…”

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confidence: 98%

Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

Wang¹,

Zhou²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

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“…In [9] non-autonomous exponential attractors were constructed for discrete time evolution processes based on the notion of uniform attractors. Subsequently, the method has been generalised for continuous time processes applying the weaker concept of pullback attractors (see [4], [5], [6], [12]).…”

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confidence: 99%

Pullback exponential attractors for differential equations with variable delays

Hammami¹,

Mchiri²,

Netchaoui³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - B

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We show how recent existence results for pullback exponential attractors can be applied to non-autonomous delay differential equations with time-varying delays. Moreover, we derive explicit estimates for the fractal dimension of the attractors. As a special case, autonomous delay differential equations are also discussed, where our results improve previously obtained bounds for the fractal dimension of exponential attractors.

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Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications

Cited by 29 publications

References 17 publications

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

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

Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

Pullback exponential attractors for differential equations with variable delays

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