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Zhu, Kaixuan; Zhou, Feng

doi:10.1016/j.camwa.2016.04.004

Computers & Mathematics with Applications

2016

DOI: 10.1016/j.camwa.2016.04.004

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Continuity and pullback attractors for a non-autonomous reaction–diffusion equation inRN

Kaixuan Zhu

Feng Zhou

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Cited by 13 publications

(11 citation statements)

References 24 publications

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“…Howbeit, there are two problem needed to be overcome. The first one is that the lack of the high-order integrability of the difference of solutions because of the cross interaction of variables in this system, unlike the single reaction-diffusion equation [5,47]. In fact, we can only obtain at most the p-times integrability of the difference of the first component of solutions as in [43].…”

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confidence: 98%

Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises

Zhao¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this article, we study the dynamical behaviour of solutions of the non-autonomous stochastic Fitzhugh-Nagumo system on R N with both multiplicative noises and non-autonomous forces, where the nonlinearity is a polynomial-like growth function of arbitrary order. An asymptotic smoothing effect of this system is demonstrated, namely, that the random pullback attractor in the initial space L 2 (R N) × L 2 (R N) is actually a compact, measurable and attracting set in H 1 (R N) × L 2 (R N). A difference estimates method, rather than the usual truncation estimate and spectrum decomposition technique, is employed to overcome the lack of Sobolev compact embedding in H 1 (R N) × L 2 (R N), despite of the loss of the high-order integrability of the difference of solutions for this system.

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mentioning

confidence: 98%

Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises

Zhao¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…Then in 2015, a Moser iteration was used by Cao, Sun & Yang [3] where the time-derivatives were avoided and the (H 1 0 ∩ L p , H 1 0 )-continuity result was proved for the stochastic system with additive Brownian noise. This technique was then further improved by Zhu & Zhou [23] in a deterministic and unbounded domain case by which the continuity result of the reaction-diffusion equation was improved to a much stronger (L 2 , H 1 0 )-continuity. However, since the analysis of [3,23] relies so heavily on interpolation inequalities, the analysis there is only for dimension N 3 and does not apply directly to all N 1, especially in unbounded domains.…”

mentioning

confidence: 99%

“…This technique was then further improved by Zhu & Zhou [23] in a deterministic and unbounded domain case by which the continuity result of the reaction-diffusion equation was improved to a much stronger (L 2 , H 1 0 )-continuity. However, since the analysis of [3,23] relies so heavily on interpolation inequalities, the analysis there is only for dimension N 3 and does not apply directly to all N 1, especially in unbounded domains. The restriction on space dimension is a natural cost of interpolation inequalities.…”

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confidence: 99%

Strong <inline-formula><tex-math id="M1">$ (L^2,L^\gamma\cap H_0^1) $</tex-math></inline-formula>-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension

Cui

Kloeden

Zhao

2020

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<p style='text-indent:20px;'>In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary <inline-formula><tex-math id="M2">$ p>2 $</tex-math></inline-formula> order nonlinearity and in any space dimension <inline-formula><tex-math id="M3">$ N \geqslant 1 $</tex-math></inline-formula>. It is proved that the weak solutions can be <inline-formula><tex-math id="M4">$ (L^2, L^\gamma\cap H_0^1) $</tex-math></inline-formula>-continuous in initial data for arbitrarily large <inline-formula><tex-math id="M5">$ \gamma \geqslant 2 $</tex-math></inline-formula> (independent of the physical parameters of the system), i.e., can converge in the norm of any <inline-formula><tex-math id="M6">$ L^\gamma\cap H_0^1 $</tex-math></inline-formula> as the corresponding initial values converge in <inline-formula><tex-math id="M7">$ L^2 $</tex-math></inline-formula>. In fact, the system is shown to be <inline-formula><tex-math id="M8">$ (L^2, L^\gamma\cap H_0^1) $</tex-math></inline-formula>-smoothing in a H<inline-formula><tex-math id="M9">$ \ddot{\rm o} $</tex-math></inline-formula>lder way. Applying this to the global attractor we find that, with external forcing only in <inline-formula><tex-math id="M10">$ L^2 $</tex-math></inline-formula>, the attractor <inline-formula><tex-math id="M11">$ \mathscr{A} $</tex-math></inline-formula> attracts bounded subsets of <inline-formula><tex-math id="M12">$ L^2 $</tex-math></inline-formula> in the norm of any <inline-formula><tex-math id="M13">$ L^\gamma\cap H_0^1 $</tex-math></inline-formula>, and that every translation set <inline-formula><tex-math id="M14">$ \mathscr{A}-z_0 $</tex-math></inline-formula> of <inline-formula><tex-math id="M15">$ \mathscr{A} $</tex-math></inline-formula> for any <inline-formula><tex-math id="M16">$ z_0\in \mathscr{A} $</tex-math></inline-formula> is a finite dimensional compact subset of <inline-formula><tex-math id="M17">$ L^\gamma\cap H_0^1 $</tex-math></inline-formula>. The main technique we employ is a combination of a Moser iteration and a decomposition of the nonlinearity, by which the interpolation inequalities are avoided and the new continuity result is obtained without any restrictions on the order <inline-formula><tex-math id="M18">$ p>2 $</tex-math></inline-formula> of the nonlinearity and the space dimension <inline-formula><tex-math id="M19">$ N \geqslant 1 $</tex-math></inline-formula>.</p>

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“…On the other hand, we overcome the difficulty caused by the fractional power Laplacian to obtain a semiintervals estimate of the bound of the solution in L 2p−2 loc (R; L 2p−2 (R N )), and then an appropriate multiplier (r − t+τ 2 ) 2p p−1 is chosen to prove the continuity of solutions in H s (R N ). This approach was used to prove the asymptotical compactness of solutions of reaction-diffusion equation, p-Laplacian equation and Fitzhugh-Nagumo system on bounded or unbounded domain, see [7,53,54,55,59].…”

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

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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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Continuity and pullback attractors for a non-autonomous reaction–diffusion equation inRN

Cited by 13 publications

References 24 publications

Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises

Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises

Strong <inline-formula><tex-math id="M1">$ (L^2,L^\gamma\cap H_0^1) $</tex-math></inline-formula>-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension

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

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