Asymptotically autonomous dynamics for parabolic equations

Li, Yangrong; She, Lianbing; Wang, Renhai

doi:10.1016/j.jmaa.2017.11.033

Cited by 42 publications

(35 citation statements)

References 22 publications

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“…In the deterministic case, some abstract criteria on backward compactness of a pullback attractor had been established by [7,25,37]. Other time-related properties of a deterministic attractor had been considered in [12,13,16,17,18,19,24].…”

Section: Renhai Wang and Yangrong LImentioning

confidence: 99%

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

Wang¹,

Li²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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We study some time-related properties of the random attractor for the stochastic wave equation on an unbounded domain with time-varying coefficient and force. We assume that the coefficient is bounded and the timedependent force is backward tempered, backward complement-small, backward tail-small, and then prove both existence and backward compactness of a random attractor on the universe of all backward tempered sets. By using the Egoroff and Lusin theorems, we show the measurability of the absorbing set although it is the union of some random sets over an uncountable index set. Moreover, we obtain the backward compactness of the attractor if the force is periodic, and obtain the periodicity of the attractor if both force and coefficient are periodic.

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Section: Renhai Wang and Yangrong LImentioning

confidence: 99%

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

Wang¹,

Li²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…Indeed, we first take an N ∈ N such that t n ≥ T for all n ≥ N, where T > 0 is the absorbing time given in (ii) of Lemma 2. We then infer from (20) and (48) in Lemma 4 that for all n ≥ N and − ≤ 1 < 2 ≤ 0,…”

Section: Definitionmentioning

confidence: 95%

“…If we take n ≡ , then the above concept reduces to the -pullback asymptotic compactness. Now, we come back to consider the continuous cocycle (defined by (20)) over Y = X for each ∈ (0, 0 ]. Lemma 6.…”

Section: Definitionmentioning

confidence: 99%

“…As a wide class of quasi-linear equations, the p-Laplacian equation is an important mathematical model for the diffusion phenomena in the non-Newtonian fluids, porous media, nonlinear elasticity, and nonlinear filtration or distribution of magnetic fields. In the nondelayed case, the long-term behavior of the p-Laplacian equation has been well investigated in the deterministic case 8,[19][20][21] and in the stochastic case. 2,5,6,[22][23][24][25][26] It is worth pointing out that there is no paper studying the stochastic delayed p-Laplacian equation, although other delayed equations had been well investigated.…”

Section: Introductionmentioning

confidence: 99%

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Double stabilities of pullback random attractors for stochastic delayed p‐Laplacian equations

Zhang

2020

Math Methods in App Sciences

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We provide a method to study the double stabilities of a pullback random attractor (PRA) generated from a stochastic partial differential equation (PDE) with delays, such a PRA is actually a family of compact random sets Aϱ(t,·), where t is the current time and ϱ is the memory time. We study its longtime stability, which means the attractor semiconverges to a compact set as the current time tends to minus infinity, and also its zero‐memory stability, which means the delayed attractor semiconverges to the nondelayed attractor as the memory time tends to zero. The stochastic nonautonomous p‐Laplacian equation with variable delays on an unbounded domain will be applied to illustrate the method and some suitable assumptions about the nonlinearity and time‐dependent delayed forces can ensure existence, backward compactness, and double stabilities of a PRA.

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“…Most recently, Li et al [10] showed that Theorem 1.1 ( [10]). Let A be the pullback attractor of a process U and A the global attractor of a semigroup S. Suppose that…”

Section: Introductionmentioning

confidence: 99%

Convergences of asymptotically autonomous pullback attractors towards semigroup attractors

Cui¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

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For pullback attractors of asymptotically autonomous dynamical systems we study the convergences of their components towards the global attractors of the limiting semigroups. We use some conditions of uniform boundedness of pullback attractors, instead of uniform compactness conditions used in the literature. Both forward convergence and backward convergence are studied.2000 Mathematics Subject Classification. 35B40, 35B41, 37L30.

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Asymptotically autonomous dynamics for parabolic equations

Cited by 42 publications

References 22 publications

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

Double stabilities of pullback random attractors for stochastic delayed p‐Laplacian equations

Convergences of asymptotically autonomous pullback attractors towards semigroup attractors

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