A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations

Li, Yangrong; Yin, Jinyan

doi:10.3934/dcdsb.2016.21.1203

Cited by 67 publications

(44 citation statements)

References 36 publications

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“…Lemma 4.5. Let (25)- (29) hold and D be defined by (38). Then the random cocycle ϕ defined by (37) is (L 2 (R N ), L 2 (R N ))-pullback asymptotically compact, namely, for every τ ∈ R and ω ∈ Ω, the sequence {ϕ(t n , τ −t n , ϑ −tn ω, u 0,n )} ∞ n=1 has a convergent subsequence in L 2 (R N ) whenever t n → ∞ and u 0,n ∈ D(τ − t n , ϑ −tn ω) with D ∈ D.…”

Section: Asymptotical Compactnessmentioning

confidence: 99%

“…According to Lemma 5.4 and (94), we immediately get Theorem 5.5. Under the conditions (25)- (29), the solution u of problem (23) and (24) is bi-spatial (L 2 (R N ), L p (R N ))-continuous with respect to the initial data. Namely, for each fixed τ ∈ R, ω ∈ Ω, the solution u(t, τ, ω, .)…”

Section: Continuity Of Solutionsmentioning

confidence: 99%

“…Theorem 5.9. Under the conditions (25)- (29), the solution u of problem (23) and (24) is bi-spatial (L 2 (R N ), H s (R N ))-continuous with respect to the initial data.…”

Section: 2mentioning

confidence: 99%

See 2 more Smart Citations

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

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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...

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Section: Asymptotical Compactnessmentioning

confidence: 99%

Section: Continuity Of Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

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

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“…The wave equation without the dispersive term was also discussed in Wang [21] and Yang, Duan and Kloeden [24] for such additive noise and in Wang, Zhou and Gu [22] for usual multiplicative noise, i.e. Su = u, also see [8,9,11,13,17,18,20,25,26,32].…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Robustness for Dispersive-Dissipative Wave Equations Driven by Small Laplace-Multiplier Noise

Wang¹,

Li²,

Li³

2018

DSA

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This paper is devoted to limit-dynamics for dispersive-dissipative wave equations on an unbounded domain. An interesting feature is that the stochastic term is multiplied by an unbounded Laplace operator. A random attractor in the Sobolev space is obtained when the density of noise is small and the growth rate of nonlinearity is subcritical. The random attractor is upper semicontinuous to the global attractor when the density of noise tends to zero. Both methods of spectrum and tail-estimate are combined to prove the collective limit-set compactness. Furthermore, a probabilistic method is used to show that the robustness of attractors is basically uniform in probability.

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“…All those assumptions are different from dealing with the pullback dynamics for other nonautonomous dissipative equations (see, e.g., [17,[21][22][23][24][25][26][27][28]). …”

Section: Introductionmentioning

confidence: 99%

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

She

Wang

2018

Discrete Dynamics in Nature and Society

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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.

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A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations

Cited by 67 publications

References 36 publications

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

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

Probabilistic Robustness for Dispersive-Dissipative Wave Equations Driven by Small Laplace-Multiplier Noise

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

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