Evolution Inclusions and Variation Inequalities for Earth Data Processing III

Zgurovsky, Michael Z.; Kasyanov, Pavlo O.; Kapustyan, O. V.; Valero, José; Zadoianchuk, Nina V.

doi:10.1007/978-3-642-28512-7

Cited by 56 publications

(57 citation statements)

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“…In the autonomous case, when A(t, y) does not depend on t, the long-time behavior of all globally defined weak solutions for Problem (13.1) is described by using trajectory and global attractors theory; Kasyanov [15,16], Zgurovsky et al [29,Chap. 2] and references therein; see also Balibrea et al [2].…”

Section: Uniform Trajectory Attractor and Main Resultsmentioning

confidence: 99%

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Uniform Trajectory Attractors for Nonautonomous Dissipative Dynamical Systems

Zgurovsky

Kasyanov

2015

Studies in Systems, Decision and Control

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For all global weak solutions of the general classes of nonautonomous evolution equations and inclusions that satisfy standard sign and polynomial growth conditions, the multivalued dynamics as time t → +∞ is studied. The existence of a compact uniform trajectory attractor is justified. The obtained results allow to investigate long-time behavior of distributions of state functions for various mathematical models in geophysics, mechanics, biology, medicine etc.

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Section: Uniform Trajectory Attractor and Main Resultsmentioning

confidence: 99%

“…Let us indicate a problem which is one of the motivations for the study of the nonautonomous evolution inclusion (13.1) (see, for example, Migórski and Ochal [22]; Zgurovsky et al [29] and references therein). In a subset Ω of R 3 ,, we consider the nonstationary heat conduction equation…”

Section: Introduction and Setting Of The Problemmentioning

confidence: 99%

Uniform Trajectory Attractors for Nonautonomous Dissipative Dynamical Systems

Zgurovsky

Kasyanov

2015

Studies in Systems, Decision and Control

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“…Now, we consider the problem It is not difficult to see that K satisfies (K1)-(K3) (see [22,Lemma 5.8]). Let us recall the following results.…”

Section: Application To a Reaction-diffusion Problemmentioning

confidence: 99%

“…Theorem 20 [11,Theorem 10] (see also [9] or [22]) The set K generates a strict multivalued semiflow G : R + × L 2 (Ω) → P (L 2 (Ω)), which possesses a global compact invariant attractor A K .…”

Section: Application To a Reaction-diffusion Problemmentioning

confidence: 99%

“…A similar situation appears for reaction-diffusion equations when we consider solutions which are regular in every interval of time of the type [ε, T ] with ε > 0. The theory of such non-strict multivalued semiflows and their attractors is developed in the paper Melnik and Valero [14], see also the monographs Kapustyan et al [9], Zgurovsky et al [22] and [10,12,15].…”

Section: Introductionmentioning

confidence: 99%

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The Envelope Attractor of Non-strict Multivalued Dynamical Systems with Application to the 3D Navier–Stokes and Reaction–Diffusion Equations

Kloeden

Marín-Rubio

Valero

2013

Set-Valued Var. Anal

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Multivalued semiflows generated by evolution equations without uniqueness sometimes satisfy a semigroup set inclusion rather than equality because, for example, the concatentation of solutions satisfying an energy inequality almost everywhere may not satisfy the energy inequality at the joining time. Such multivalued semiflows are said to be non-strict and their attractors need only be negatively semi-invariant. In this paper the problem of enveloping a non-strict multivalued dynamical system in a strict one is analyzed and their attactors are compared. Two constructions are proposed. In the first, the attainability set mapping is extending successively to be strict at the dyadic numbers, which essentially means (in the case of the Navier-Stokes system) that the energy inequality is satisfied piecewise on successively finer dyadic subintervals. The other deals directly with trajectories and their concatenations, which are then used to define a strict multivalued dynamical system. The first is shown to be applicable to the three-dimensional Navier-Stokes equations and the second to a reaction-diffusion problem without unique solutions.

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