The Attractor for a Nonlinear Reaction-Diffusion System in the Unbounded Domain and Kolmogorove's ɛ-Entropy

Zelik, Sergey

doi:10.1002/1522-2616(200112)232:1<129::aid-mana129>3.0.co;2-t

Cited by 37 publications

(7 citation statements)

References 16 publications

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“…The intention of this article is to show the existence of a compact positively invariant set A * which exponentially attracts every bounded set for multi-valued semidynamical systems. It is worth mentioning that here we do not consider the finite dimensionality of the set A * , since it is difficult to show that multi-valued systems possess some kind of smoothing property, which was used in the construction of the exponential attractors for the single-valued case, see, e.g., [2,12,13,14,15,24,34], and in fact, many single-valued semigroups have infinite dimensional global attractors, see, e.g., [33,36].…”

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Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

Wang¹,

Lin²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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We first prove the existence of a compact positively invariant set which exponentially attracts any bounded set for abstract multi-valued semidynamical systems. Then, we apply the abstract theory to handle retarded ordinary differential equations and lattice dynamical systems, as well as reactiondiffusion equations with infinite delays. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.

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Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

Wang¹,

Lin²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…For reaction-diffusion equations u t = ∆u + u − u 3 , the solutions u = ±1 and the traveling wave connections between u = 0 and u = ±1 are no longer included the Sobolev spaces like L p1 (R N )(1 ≤ p 1 < ∞), for example, see [23]. Hence, in [3,4,8,20,[41][42][43], the authors introduced locally uniform spaces as the phase space to include these special solutions in the global attractors. The idea of the locally uniform spaces can be traced back to Kato [24].…”

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“…In section 5, we consider the global well-posedness of problem (1.1)-(1.2) in H 2(α− ),q U (R N ) by a comparison principle. Motivated by ideas of [15,20,41,43](see also, for instance, [40,42,44]), we obtain continuous property of a family of process in H 2(α− ),q φ (R N ) by extension solutions obtained in section 4. In section 6, we prove Theorem 1.1.…”

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Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces

Yue¹

2017

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we first prove the well-posedness for the nonautonomous reaction-diffusion equations with fractional diffusion in the locally uniform spaces framework. Under very minimal assumptions, then we study the asymptotic behavior of solutions of such equation and show the existence of (Hwith locally uniform external forces being translation uniform bounded but not translation compact in L p b (R; L q U (R N )). The key to that extensions is a new the space-time estimates in locally uniform spaces for the linear fractional power dissipative equation. N 2 Γ(1 − α)). In the limits s → 0 and 2010 Mathematics Subject Classification. Primary: 35K57, 35B40; Secondary: 35B41.

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Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

Zelik¹

2003

Comm Pure Appl Math

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The nonlinear reaction-diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor A in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ε-entropy. Upper and lower bounds of this entropy are obtained.Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in R n . In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional "time" if n > 1) acting on a phase space A. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy.In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz-continuous) embedding of this system into the spatial shifts on the attractor.Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution.

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The Attractor for a Nonlinear Reaction-Diffusion System in the Unbounded Domain and Kolmogorove's ɛ-Entropy

Cited by 37 publications

References 16 publications

Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

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