Attractors of partial differential evolution equations in an unbounded domain

Babin, Anatoli; Vishik, M. I.

doi:10.1017/s0308210500031498

Cited by 198 publications

(198 citation statements)

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“…Note that several different versions of (1.1), including the case of systems, have been considered by a number of authors, e.g. [1,6,9,12,13,18,19,23,27,32,33,34], where attractors have been constructed under suitable functional settings and suitable growth sign and structure conditions on the nonlinear term f (x, u, ∇u).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For example, weighted Sobolev spaces W k,p ρ R N have been used in [1,9,18,21,22]. The space of bounded and uniformly continuous functions BU C R N , was used in [32].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For example in [9,14] the assumption on the nonlinear term reads f (x, u) = −λ 0 u + f 0 (u) + g(x), with λ 0 > 0, f 0 (u)u ≤ 0, (…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Dissipative parabolic equations in locally uniform spaces

Arrieta

Cholewa

Dlotko

et al. 2007

Mathematische Nachrichten

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Keywords Reaction-diffusion equations, Cauchy problem in R N ,dissipativeness, global attractor MSC (2000) 35K15, 35K57, 35B40, 35B41TheCauchyproblem forasemilinear second orderparabolic equation The Cauchy problem for a semilinear second order parabolic equationN , is considered within the semigroup approach in locally uniform spacesẆGlobal solvability, dissipativeness and the existence of an attractor are established under the same assumptions as for problems in bounded domains. In particular, the condition sf (s, 0) < 0, |s| > s0 > 0, together with gradient's "subquadratic" growth restriction, are shown to guarantee the existence of an attractor for the above mentioned equation. This result cannot be located in the previous references devoted to reaction-diffusion equations in the whole of R N .

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For example, weighted Sobolev spaces W k,p ρ R N have been used in [1,9,18,21,22]. The space of bounded and uniformly continuous functions BU C R N , was used in [32].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Dissipative parabolic equations in locally uniform spaces

Arrieta

Cholewa

Dlotko

et al. 2007

Mathematische Nachrichten

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“…The operator S t (u 0 ) : H → H generated by (6.4) is the differential of S t on A at the point u 0 (see [6]). We then apply Theorem 3.2 of [6].…”

Section: Lemma 53 Operators {S T } Are Continuous From H(γ) Into H(γ)mentioning

confidence: 99%

“…Such properties are no longer valid for equations on unbounded domains, and it is thus more difficult to develop a general theory of existence of the global attractor in this case. A possibility then consists of working in weighted Sobolev spaces (see [1], [3], [4], [6], and [10]). …”

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confidence: 99%

Finite-dimensional attractor for the viscous Cahn-Hilliard equation in an unbounded domain

Bonfoh

2006

Quart. Appl. Math.

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Abstract.We consider the viscous Cahn-Hilliard equation in an infinite domain. Due to the noncompactness of operators, we use weighted Sobolev spaces to prove that the semigroup generated by this equation has the global attractor which has finite Hausdorff dimension.1. Introduction. Many equations arising from mechanics and physics possess a global attractor, which is a compact invariant set which uniformly attracts the trajectories as time goes to infinity, and thus appears as a suitable object for the study of the asymptotic behaviour of the system. An important issue is then to study the dimension, in the sense of the Hausdorff or fractal dimension, of the global attractor. A finite bound of the dimension of the attractor means that the system has an asymptotic behaviour determined by a finite number of degrees of freedom, indeed a remarkable improvement compared to the a priori infinite-dimensional dynamics (see [5] and [15]).For the equations on bounded domains, the known constructions of global attractors make use of some compactness properties in an essential manner, and more specifically of the compact embedding of H m 1 into H m 2 , when m 1 > m 2 . Such properties are no longer valid for equations on unbounded domains, and it is thus more difficult to develop a general theory of existence of the global attractor in this case. A possibility then consists of working in weighted Sobolev spaces (see In this article, we study, on an unbounded domain, the existence of global attractors and their Hausdorff dimensions for the viscous Cahn-Hilliard equation of the form

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Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: An example

Chepyzhov

Efendiev

2000

Comm Pure Appl Math

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The subject of this paper is the asymptotic behavior of a class of nonautonomous, infinite‐dimensional dynamical systems with an underlying unbounded domain. We present an approach that is able to overcome both the law of compactness of the trajectories and the continuity of the spectrum of the linear part of the equations under consideration, providing nevertheless existence of uniform attractors. Moreover, our approach allows us to estimate the Hausdorff dimension of attractors of nonautonomous equations in terms of physical parameters. © 2000 John Wiley & Sons, Inc.

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Attractors of partial differential evolution equations in an unbounded domain

Cited by 198 publications

References 4 publications

Dissipative parabolic equations in locally uniform spaces

Dissipative parabolic equations in locally uniform spaces

Finite-dimensional attractor for the viscous Cahn-Hilliard equation in an unbounded domain

Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: An example

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