On the Existence of the Compact Global Attractor for Semilinear Reaction Diffusion Systems on RN

Merino, Sandro

doi:10.1006/jdeq.1996.0172

Cited by 42 publications

(34 citation statements)

References 15 publications

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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]. More recently Bessel potentials spaces H s p R N have been used in [6].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Slightly weaker assumptions on the linear part are used in [32] and even weaker ones have been considered in [6]. On the other hand assuming monotonicity conditions on f 0 , conditions on the linear term are dropped in [13,22], which implies that the bistable equation can be handled within these frameworks.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the last fifteen years a number of problems in unbounded domains originating in applications have been intensively studied [1,6,9,12,13,14,15,16,18,19,20,23,32,33,34,45]. The references contain merely a sample list of the related publications.…”

Section: Comparison With Known Results and Examplesmentioning

confidence: 99%

“…This is also the reason why the BU C setting is somewhat easier for local existence than the L ∞ setting, see e.g. [32]. On the other hand for the case of weighted spaces, although the linear semigroup is continuous (see [7]), the difficulties connected with the nonlinear equations come from the lack of Sobolev-like embeddings which make unavailable the standard techniques for handling nonlinear terms.…”

Section: Wwwmn-journalcommentioning

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%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Comparison With Known Results and Examplesmentioning

confidence: 99%

Section: Wwwmn-journalcommentioning

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

Zelik

2001

Math. Nachr.

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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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On the Existence of the Compact Global Attractor for Semilinear Reaction Diffusion Systems on RN

Cited by 42 publications

References 15 publications

Dissipative parabolic equations in locally uniform spaces

Dissipative parabolic equations in locally uniform spaces

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

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

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