Construction of nonautonomous forward attractors

Kloeden, Peter E.; Lorenz, Thomas

doi:10.1090/proc/12735

Cited by 49 publications

(56 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, under Assumptions A1 -A2 , it is possible to prove the existence of a forward attractor (see [32,33]), which may be different from the pullback attractor.…”

Section: Remark 15 (I)mentioning

confidence: 99%

See 1 more Smart Citation

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

She

Wang

2018

Discrete Dynamics in Nature and Society

View full text Add to dashboard Cite

This paper deals with pullback dynamics for the weakly damped Schrödinger equation with time-dependent forcing. An increasing, bounded, and pullback absorbing set is obtained if the forcing and its time-derivative are backward uniformly integrable. Also, we obtain the forward absorption, which is only used to deduce the backward compact-decay decomposition according to high and low frequencies. Based on a new existence theorem of a backward compact pullback attractor, we show that the nonautonomous Schrödinger equation has a pullback attractor which is compact in the past. The method of energy, high-low frequency decomposition, Sobolev embedding, and interpolation are quite involved in calculating a priori pullback or forward bound.

show abstract

“…In fact, under Assumptions A1 -A2 , it is possible to prove the existence of a forward attractor (see [32,33]), which may be different from the pullback attractor.…”

Section: Remark 15 (I)mentioning

confidence: 99%

“…It may be possible to deduce a forward attractor (cf. [32,33]), but we do not pursue this forward attractor in the present paper.…”

Section: Introductionmentioning

confidence: 99%

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

She

Wang

2018

Discrete Dynamics in Nature and Society

View full text Add to dashboard Cite

show abstract

“…The important observation is that the expression (9) holds inside any positively invariant family of sets [47], regardless of what is happening outside it. Moreover, a forward attractor is always contained in such a positively invariant family [43,44].…”

Section: Existence Of Forward Attractors For Processesmentioning

confidence: 99%

“…The expression (9) gives subsets A(t) = {0}, which are in fact the component subsets of a pullback attractor, but not a forward attractor as the forward ω-limit set is [−1, 1], see [45]. Conditions excluding this case are discussed in [43,44].…”

Section: Existence Of Forward Attractors For Processesmentioning

confidence: 99%

Nonautonomous and Random Attractors

Crauel

Kloeden

2015

Jahresber. Dtsch. Math. Ver.

Self Cite

View full text Add to dashboard Cite

The theories of nonautonomous and random dynamical systems have undergone extensive, often parallel, developments in the past two decades. In particular, new concepts of nonautonomous and random attractors have been introduced. These consist of families of sets that are mapped onto each other as time evolves and have two forms: a forward attractor based on information about the system in the future and a pullback attractor that uses information about the past of the system. Both reduce to the usual attractor consisting of a single set in the autonomous case.Keywords Skew product flow · 2-parameter semi-group · Pullback attractor · Forward attractor · Random dynamical system · Weak attractor · Mean-square random dynamical system · Mean-field stochastic differential equations Mathematics Subject Classification (2010) 34D45 · 35B41 · 37-02 · 37B55 · 37C70 · 37H99 · 37L30 · 37L55 · 60H10 · 60H15

show abstract

“…As for the attractor with forward attraction of point-wise convergence is considered by Kloeden [16,23,24,25]. However, the considered convergence in this paper is in the sense of pullback attraction, which is stronger than the forward attraction in probability, defined by (2).…”

mentioning

confidence: 99%

Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain

Zhao¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

In this article, a notion of bi-spatial continuous random dynamical system is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptotically compact and random absorbing in the initial space, then it admits a bispatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the nonautonomous stochastic fractional power dissipative equation on R N with additive white noise and a polynomial-like growth nonlinearity of order p, p ≥ 2. We prove that this equation generates a bi-spatial (L 2 (R N ), H s (R N ) ∩ L p (R N ))continuous random dynamical system, and the random dynamics for this system is captured by a bi-spatial pullback attractor which is compact and attracting in H s (R N ) ∩ L p (R N ), where H s (R N ) is a fractional Sobolev space with s ∈ (0, 1). Especially, the measurability of pullback attractor is individually derived by proving the the continuity of solutions in H s (R N ) and L p (R N ) with respect to the sample. A difference estimates approach, rather than the usual truncation estimate and spectral decomposition technique, is employed to overcome the loss of Sobolev compact embedding in H s (R N ) ∩ L p (R N ), s ∈ (0, 1), N ≥ 1.2010 Mathematics Subject Classification. Primary: 35R60, 35B40, 35B41; Secondary: 35B65. Key words and phrases. Bi-spatial continuous random dynamical system, fractional power dissipative equation, fractional Sobolev space, pullback attractor, measurability.The operator (−∆) s with s ∈ (0, 1) is called a fractional power Laplacian, whose limit as s ↑ 1 is the classic Laplacian ∆, see [17]. It is a nonlocal generalization of the classic Laplacian that is often used to model the diffusive processes, see [17,27,40] and the references therein. As for the long-time dynamics, Hu et al [32] discussed the existence of a random attractor in L 2 (R N ) for the fractional power dissipative stochastic equation with additive noises. Wang [46] discussed the existence and uniqueness of random attractor for the same equation with multiplicative noise on bounded domain. The dynamics of 3D Ginzburg-Landau equation involving fractional power Laplacian ware studied in [33,34]. Most recently, Gu et al [21] obtained the regularity of the random attractor for problem (1) in H s (R N ) in the case of multiplicative noise, where the authors employed the tail estimate and spectral decomposition technique to overcome the loss of compactness on unbounded domains. But little is known about the continuity and measurability of its solutions in H s (R N ) and L p (R N ) for s ∈ (0, 1) and p ≥ 2. In this paper, we prove the existence, regularity and measurability of pullback attractors for the stochastic fractional power dissipative equation (1) defined on the whole space R N .Most of the stochastic partial differential equations (SPDEs) present a regular solut...

show abstract

Construction of nonautonomous forward attractors

Cited by 49 publications

References 18 publications

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

Nonautonomous and Random Attractors

Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain

Contact Info

Product

Resources

About