Pullback Asymptotic Compactness Research Articles

Existence of time-dependent attractor of wave equation in locally uniform space

In this article, we study non-autonomous dynamical behavior of weakly damped wave equation in unbounded domain. First of all, we introduce the time-dependent locally uniform space. After that, the pullback asymptotical compactness is proved by applying the contractive function method. Eventually, we obtain the existence of ( H lu 1 ( R 3 ) × L lu 2 ( R 3 ) , H ρ 1 ( R 3 ) × L ρ 1 ( R 3 ) ) -time-dependent attractor of wave equation.

Asymptotically Autonomous Robustness of Non-autonomous Random Attractors for Stochastic Convective Brinkman-Forchheimer Equations on ℝ3

Abstract This article is concerned with the asymptotically autonomous robustness (almost surely and in probability) of random attractors for stochastic version of 3D convective Brinkman-Forchheimer (CBF) equations defined on $\mathbb {R}^{3}$: $$ \begin{align*} &\frac{\partial\boldsymbol{v}}{\partial\mathrm{t}}-\mu\Delta\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+\alpha\boldsymbol{v}+\beta|\boldsymbol{v}|^{r-1}\boldsymbol{v}+\nabla{p}=\boldsymbol{f}+``\mbox{stochastic terms}",\quad\nabla\cdot\boldsymbol{v}=0,\end{align*}$$where $\mu ,\alpha ,\beta > 0$, $r\geq 1$ and $\boldsymbol {f}(\cdot )$ is a given time-dependent external force field. Our goal is to study the asymptotically autonomous robustness for 3D stochastic CBF equations perturbed by a linear multiplicative or additive noise when time-dependent forcing converges towards a time-independent function. The main procedure to achieve our goal is how to justify that the usual pullback asymptotic compactness of the solution operators is uniform on some uniformly tempered universes over an infinite time-interval $(-\infty ,\tau ]$. This can be done by showing the backward uniform “tail-smallness” and “flattening-property” of the solutions over $(-\infty ,\tau ]$.

Random attractors for a stochastic nonlocal delayed reaction–diffusion equation on a semi-infinite interval

Abstract The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.

Necessary and sufficient criteria for existence, regularity, and asymptotic stability of enhanced pullback attractors with applications to 3D primitive equations

We introduce several new concepts called enhanced pullback attractors for nonautonomous dynamical systems by improving the compactness and attraction of the usual pullback attractors in strong topology spaces uniformly over some infinite time intervals. Then we establish several necessary and sufficient criteria for the existence, regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which are expected to be applied to many kinds of nonautonomous PDEs. These abstract results are applied to the famous 3D primitive equations modeling the large-scale ocean and atmosphere dynamics. We establish the existence, regularity and asymptotic stability of the enhanced pullback attractors for the nonautonomous primitive equations in [Formula: see text] and [Formula: see text] when the forces satisfy some mild conditions. The uniform pullback asymptotic compactness of the solution operators in [Formula: see text] and [Formula: see text] are established by deriving the uniform “flattening effects” of the solutions in [Formula: see text] and the uniform [Formula: see text]-Hölder continuity of the solutions from [Formula: see text] to [Formula: see text] with respect to the initial data, respectively. Our approaches are applicable to prove the existence of global attractors for the autonomous primitive equations in [Formula: see text] provided [Formula: see text].

Asymptotic behavior for stochastic plate equations on unbounded domains

In this paper, we investigate the long-time behavior of the solutions for stochastic plate equations with memory and additive noise. First we establish a continuous cocycle for the equation and the pullback asymptotic compactness of solutions. Second we prove the existence of random attractors for the equation.

Existence and upper semicontinuity of random pullback attractors for 2D and 3D non-autonomous stochastic convective Brinkman-Forchheimer equations on whole space

In this work, we analyze the long time behavior of 2D as well as 3D convective Brinkman-Forchheimer (CBF) equations and its stochastic counterpart with non-autonomous deterministic forcing term in $\mathbb{R}^d$ $ (d=2, 3)$: $$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u} + (\boldsymbol{u}\cdot\nabla)\boldsymbol{u} + \alpha\boldsymbol{u} + \beta|\boldsymbol{u}|^{r-1} \boldsymbol{u} + \nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0, $$ where $r\geq1$. We prove the existence of a unique global pullback attractor for non-autonomous CBF equations, for $d=2$ with $r\geq1$ and $d=3$ with $r > 3$ (for any $\beta,\mu > 0$), and $d=r=3$ with $2\beta\mu\geq1$. For the same cases, we show the existence of a unique random pullback attractor for non-autonomous stochastic CBF equations with linear multiplicative white noise. In addition, a remark on the existence of a unique weak pullback mean random attractor for the stochastic CBF equations is also provided. Finally, we establish the upper semicontinuity of the random pullback attractors, that is, the random pullback attractors converge towards the global pullback attractor when the noise intensity approaches to zero. Since we do not have compact Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of the random dynamical system is proved by the method of energy equations given by Ball. For the case of Navier-Stokes equations defined on $\mathbb{R}^d$, such results are not available and the presence of Darcy term $\alpha\boldsymbol{u}$ helps us to establish the above mentioned results for CBF equations on whole space.

Forward dynamics of 3D double time-delayed MHD-Voight equations

This paper is concerned with the long-term behaviour of 3D double time-delayed magnetohydrodynamics (MHD)-Voight equations with non-autonomous forcing term. First, we establish the existence, forward compactness and forward stability of pullback attractors by the forward equi-continuity and forward pullback asymptotic compactness of solutions. We then study the upper semi-convergence of pullback attractors as the time parameter tends to positive infinity. Since the solution of this equation has no high regularity, we use the spectrum decomposition technique to prove the forward pointwise pullback asymptotic compactness of solutions.

Dynamics for subcritical fractional nonclassical diffusion equations with nonlinear Wong-Zakai noise and delays

The multi-scale stability of random attractors in the space of continuous functions from the delay interval into the fractional Sobolev space for a class of nonclassical diffusion equations on $ \mathbb{R}^n $ with time-delay and nonlinear Wong-Zakai noise is studied. First, we prove the existence of a unique pullback random attractor, which is a family of compact sets depending on two parameters: the step size of noise and the current time. Secondly, we study the semicontinuity of the pullback random attractor as the current time goes to positive infinity. Thirdly, we consider the semicontinuity of the pullback random attractor as the delay time approach zero. Finally, we prove the semicontinuity of the pullback random attractor as the step size of noise tends to positive infinity. In order to overcome the lack of compact Sobolev embeddings on $ \mathbb{R}^n $ and the weak dissipation of equations, we employ the spectral method and idea of uniform tail-estimates to prove that the solution operator is pullback asymptotic compactness over a time-uniformly tempered attracting universe.

Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $

<p style='text-indent:20px;'>This paper deals with the asymptotic behavior of the solutions to a class of non-autonomous <i>fractional</i> stochastic <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian equation driven by linear additive noise on the entire space <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. We firstly prove the existence of a continuous non-autonomous cocycle for the equation as well as the uniform estimates of solutions. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors and the uniform tail-estimates of the solutions for large space variables when time is large enough to surmount the lack of compact Sobolev embeddings on unbounded domains. Finally, we establish the upper semi-continuity of the random attractors when noise intensity approaches zero.</p>

Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains

<abstract><p>This paper is concerned with the asymptotic behavior of the stochastic three dimensional Brinkman-Forchheimer equations in some unbounded domains. We first define a continuous random dynamical system for the equations. Then by J. Ball's idea of energy equations, we obtain pullback asymptotic compactness of solutions and prove that the existence of a unique random attractor for the equations.</p></abstract>

Asymptotic behavior of non-autonomous fractional stochastic <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian equations with colored noise and delay on <inline-formula><tex-math id="M2">$ \mathbb{R}^n $</tex-math></inline-formula>

This paper deals with the asymptotic behavior of the solutions to a class of non-autonomous fractional stochastic $ p $-Laplacian equations with delay driven by nonlinear colored noise on the entire space $ \mathbb{R}^n $. We firstly considered the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the time delay. We then showed pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform tail-estimates of the solutions for large space variables when time is large enough to surmount the lack of compact Sobolev embeddings on unbounded domains.

Asymptotic Behavior of Random Reaction-diffusion Delay Equations Driven by Colored Noise

In this paper, we consider the asymptotic behavior of random reaction-diffusion delay equations driven by colored noise defined on unbounded domains. We firstly establish the existence and uniqueness of pullback random attractors for the continuous cocycle associated with the equation, and then consider the convergence of random attractors when colored noise approximates white noise. The methods of uniform tail-estimate and operator decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embeddings on unbounded domains.

Asymptotically autonomous dynamics for fractional subcritical nonclassical diffusion equations driven by nonlinear colored noise

The asymptotically autonomous dynamics in $$H^s(\mathbb R^n)$$ for any $$s\in (0,1)$$ and $$n\in \mathbb {N}$$ are discussed for a class of highly nonlinear nonclassical diffusion equations perturbed by colored noise. The main feature of this model is that the time-dependent drift and diffusion terms have subcritical and superlinear polynomial growth of orders p and q, respectively, satisfying $$\begin{aligned} 2\le 2q<p<\infty \text { if }n=1\text { and }s\in [\frac{1}{2},1);\text { otherwise, }\, 2\le 2q<p<\frac{2n}{n-2s}. \end{aligned}$$ The number $$\frac{2n}{n-2s}$$ is called the fractional critical Sobolev embedding exponent. Under this setting, we prove the existence, time-dependent uniform compactness and asymptotically autonomous convergence of pathwise random attractors of the equations in $$H^s(\mathbb R^n)$$ when the time-dependent nonlinearities satisfy some new conditions. The time-dependent uniform pullback asymptotical compactness of the solution operators in $$H^s(\mathbb R^n)$$ is proved by virtue of a cut-off technique [38], a spectral decomposition approach and uniform estimates in a time-uniformly tempered attracting universe in order to overcome several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains, the weak dissipativeness of the systems and the unknown measurability of attractors.

Dynamical stability of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise

In this paper, two problems related to FitzHugh–Nagumo lattice systems are analyzed. The first one is concerned with the asymptotic behavior of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlation time of Wong–Zakai noise goes to infinity rather than to zero. We first prove the existence of tempered random attractors for the random delayed lattice systems with a nonlinear drift function and a nonlinear diffusion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli–Arzelà theorem and uniform tail-estimates. We then show the upper semicontinuity of attractors as the correlation time tends to infinity. As for the second problem, we consider the corresponding deterministic version of the previous model and study the convergence of attractors when the delay approaches zero. That is, the upper semicontinuity of attractors for the delayed system to the non-delayed one is proved.

Backward regularity of attractors for lattice FitzHugh-Nagumo system with double random coefficients

In this paper, we investigate the backward compact dynamical behavior for the stochastic lattice FitzHugh-Nagumo system with double random coefficients and multiplicative noise. It is proved, under some suitable conditions on the nonlinear term and the body force, that the system has a unique backward compact bi-spatial random (ℓσ2×ℓσ2,ℓσp×ℓσ2)-attractor for any p>2. By such a bi-spatial attractor we mean an invariant backward compact set in ℓσp×ℓσ2 that pullback attracts all nonempty subsets of ℓσ2×ℓσ2 under another topology of ℓσp×ℓσ2. The time-dependent uniform pullback asymptotical compactness of the solution operators in ℓσp×ℓσ2 is proved by virtue of a cut-off technique and considering two different attracting universes to overcome several difficulties caused by the lack of compact Sobolev embeddings in the case of a lattice system and the unknown measurability of random attractor.

Pullback Attractors for a Class of Semilinear Second‐Order Nonautonomous Evolution Equations with Hereditary Characteristics

In this paper, we investigate the long‐time behavior for the nonautonomous semilinear second‐order evolution equation (∂2u/∂t2) − Δu − Δ(∂u/∂t) − Δ(∂2u/∂t2) = f(t, u(x, t − ρ(t))) + g(t, x), in(τ, ∞) × Ω with some hereditary characteristics, where Ω is an open‐bounded domain of ℝN(N ≥ 3) with smooth boundary ∂Ω. Firstly, we establish the existence of solutions for the second‐order nonautonomous evolution equation by the standard Faedo–Galerkin method, but without the uniqueness of solutions. Then by proving the pullback asymptotic compactness for the multivalued process {U(t, τ)} on , we obtain the existence of pullback attractors in the Banach spaces for the multivalued process generated by a class of second‐order nonautonomous evolution equations with hereditary characteristics and ill‐posedness.

Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains

&lt;p style='text-indent:20px;'&gt;In this paper we study asymptotic behavior of a class of stochastic plate equations with memory and additive noise. First we introduce a continuous cocycle for the equation and establish the pullback asymptotic compactness of solutions. Second we consider the existence and upper semicontinuity of random attractors for the equation.&lt;/p&gt;

Stochastic dynamics of non-autonomous fractional Ginzburg-Landau equations on $ \mathbb{R}^3 $

&lt;p style='text-indent:20px;'&gt;We investigate a class of non-autonomous non-local fractional stochastic Ginzburg-Landau equation with multiplicative white noise in three spatial dimensions. Of particular interest is the asymptotic behavior of its solutions. We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ L^2( \mathbb{R}^3) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. The existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions is then established. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ L^2( \mathbb{R}^3) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables.&lt;/p&gt;

Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains

This paper deals with the long-time dynamics for a class of highly nonlinear fractional nonclassical diffusion equations with nonlinear colored noise and time delay defined on the whole space Rn. The existence and uniqueness of tempered pullback random attractors of the equations are established in C([−ρ,0],Hα(Rn)) (ρ>0 and α∈(0,1)) for polynomial growth drift and diffusion terms as well as Lipschitz time-delay terms. In a special case where the nonlinear diffusion term depends only on the space variable, the approximation of those random attractors is also investigated in C([−ρ,0],Hα(Rn)) when the correlation time of the colored approaches zero. The pullback asymptotical compactness of the solutions in C([−ρ,0],Hα(Rn)) is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates developed by Wang (1999) [59] in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains as well as the weakly dissipative distinguishing structures of the equations.

Wong-Zakai approximations and attractors for non-autonomous stochastic FitzHugh-Nagumo system on unbounded domains

In this paper, we study the Wong-Zakai approximations and long term behavior of the stochastic FitzHugh-Nagumo system driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation system. Then we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic FitzHugh-Nagumo system for both additive and multiplicative noise on unbounded domains. The tail estimates and decomposition method are employed to derive the pullback asymptotic compactness of solutions in order to overcome the obstacles caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the absence of regularity of one component of the solutions.