Lower Semicontinuity of a Non-Hyperbolic Attractor

Kostin, I. N.

doi:10.1112/jlms/52.3.568

Cited by 24 publications

(12 citation statements)

References 13 publications

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“…Let f be real analytic and satisfy (3)- (7). If ( 0 , 1 ) ∈ H 0 and ( (t), (* /*t)(t)) is the corresponding trajectory, then On the other hand, it is not difficult to realize that, on account of (125),…”

Section: Theorem 83mentioning

confidence: 98%

Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation

Bonfoh

Grasselli

Miranville

2007

Math Methods in App Sciences

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SUMMARYWe consider a singular perturbation of the generalized viscous Cahn-Hilliard equation based on constitutive equations introduced by Gurtin. This equation rules the order parameter , which represents the density of atoms, and it is given on a n-rectangle (n 3) with periodic boundary conditions. We prove the existence of a family of exponential attractors that is robust with respect to the perturbation parameter >0, as goes to 0. In a similar spirit, we analyze the stability of the global attractor. If n = 1, 2, then we also construct a family of inertial manifolds that is continuous with respect to . These results improve and generalize the ones contained in some previous papers. Finally, we establish the convergence of any trajectory to a single equilibrium via a suitable version of the Łojasiewicz-Simon inequality, provided that the potential is real analytic.

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Section: Theorem 83mentioning

confidence: 98%

Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation

Bonfoh

Grasselli

Miranville

2007

Math Methods in App Sciences

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“…One can refer to Babin and Vishik [9], Caraballo and Langa [10], Elliott and Kostin [11], Hale [2], Hale et al [12], Hale and Raugel [13,14], Kostin [15], Lu and Wang [16], Lv and Wang [17,18], Vleck and Wang [19], Zhao and Zhou [20,21], etc.…”

Section: Introductionmentioning

confidence: 99%

Upper semicontinuity of attractors for lattice systems under singular perturbations

Zhao

Zhou

2010

Nonlinear Analysis: Theory, Methods & Applications

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“…Their proof of this result relies on A. N. Carvalho et al the continuity of the equilibria and lower semicontinuity of the local unstable manifolds under perturbation (see also Stuart and Humphries [29], and Kostin [22] and Elliot and Kostin [16], who use a similar argument based on the continuity of unstable manifolds but show that this remains applicable in certain systems with non-hyperbolic equilibria when the unperturbed problem possesses a Lyapunov functional). Several applications have been considered in [1,2,5,8,9] (including more complicated situations where even the continuity of the equilibria is not straightforward), which maintain the hypothesis of a gradient structure for the limit problem.…”

Section: Introductionmentioning

confidence: 99%

“…Several applications have been considered in [1,2,5,8,9] (including more complicated situations where even the continuity of the equilibria is not straightforward), which maintain the hypothesis of a gradient structure for the limit problem. In all of these papers (with the exception of [16] and [22]), a key step was to show the uniformity (with respect to the underlying parameter) of the exponential dichotomy of the linearization around each hyperbolic equilibrium.…”

Section: Introductionmentioning

confidence: 99%

Lower semicontinuity of attractors for non-autonomous dynamical systems

Carvalho¹,

Langa²,

Robinson³

2009

Ergod. Th. Dynam. Sys.

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Abstract. This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions. IntroductionResults on the upper semicontinuity of attractors with respect to perturbations ('no explosion') are relatively easy to prove, and are now essentially classical. Results on lower semicontinuity ('no collapse') are much more difficult: generally they involve assumptions on the structure of the unperturbed attractor, and one then tries to reproduce a similar structure inside the perturbed attractor. Currently, these lower semicontinuity results are restricted to the class of autonomous dynamical systems that are gradient or 'gradient-like', i.e. systems for which the global attractor is given by the union of the unstable manifolds of a finite set of hyperbolic equilibria.The study of lower semicontinuity of attractors under perturbation, given this gradient assumption, was set in motion by Hale and Raugel [19], who proved an abstract result and considered applications to partial differential equations. Their proof of this result relies on

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Lower Semicontinuity of a Non-Hyperbolic Attractor

Abstract: The unstable invariant set near a non‐hyperbolic stationary point of an abstract parabolic equation is studied. Lower semicontinuity of the attractor for the Chafee‐Infante problem in the case of the non‐hyperbolic zero stationary solution is proved.

Cited by 24 publications

References 13 publications

Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation

Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation

Upper semicontinuity of attractors for lattice systems under singular perturbations

Lower semicontinuity of attractors for non-autonomous dynamical systems

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