Upper Semicontinuity Of Random Attractors Research Articles

Upper Semicontinuity of Random Attractors for Random Differential Equations with Nonlinear Diffusion Terms I: Finite-Dimensional Case

Upper Semicontinuity of Random Attractors for Random Differential Equations with Nonlinear Diffusion Terms I: Finite-Dimensional Case

Stability of stochastic reaction-diffusion equation under random influences in high regular spaces

In this paper, we systematically study the high-order stability of the stochastic reaction-diffusion equation driven by additive noise as the noise intensity vanishes. First, with a general assumption on the nonlinear term, we obtain the convergence of solutions and upper semi-continuity of random attractors in L2(RN). Second, by using the nonlinear decomposition method, we technically establish the convergence of solutions in Lp(RN)∩H1(RN)(p>2), and therefore, the upper semi-continuity of random attractors is proved, where p is the growth exponent of the nonlinearity. Finally, by induction argument, we prove that the solution is uniformly bounded near the initial time in Lδ(RN) for arbitrary δ > p, in which space the convergence of solutions and the upper semi-continuity of random attractors are also established.

Wong–Zakai approximations for non-autonomous stochastic parabolic equations with X-elliptic operators in higher regular spaces

In this paper, we consider the regularity of Wong–Zakai approximations of the non-autonomous stochastic degenerate parabolic equations with X-elliptic operators. We first establish the pullback random attractors for the random degenerate parabolic equations with a general diffusion. Then, we prove the convergence of solutions and the upper semi-continuity of random attractors of the Wong–Zakai approximation equations in Lp(DN) ∩ H.

Dynamics of a stochastic fractional nonlocal reaction-diffusion model driven by additive noise

In this paper, we are concerned with the long-time behavior of stochastic fractional nonlocal reaction-diffusion equations driven by additive noise. We use the techniques of random dynamical systems to transform the stochastic model into a random one. To deal with the new nonlocal term appeared in the transformed equation, we first use a generalization of Peano's theorem to prove the existence of local solutions, and then adopt the Galerkin method to prove existence and uniqueness of weak solutions. Next, the existence of pullback attractors for the equation and its associated Wong-Zakai approximation equation driven by colored noise are shown, respectively. Furthermore, we establish the upper semi-continuity of random attractors of the Wong-Zakai approximation equation as $ \delta\rightarrow0^{+} $.

Wong-Zakai approximations of stochastic lattice systems driven by long-range interactions and multiplicative white noises

<p style='text-indent:20px;'>In this paper, we study the Wong-Zakai approximations of a stochastic lattice differential equation with long-range interactions and multiplicative white noise at each node. We first prove the existence and uniqueness of pullback random attractors for lattice system driven by multiplicative white noises as well as the corresponding Wong-Zakai approximate system. Then, we prove the convergence of solutions and the upper semicontinuity of random attractors for the Wong-Zakai approximate system as the size of approximation approaches zero.</p>

Existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman–Forchheimer equations in bounded domains

In this work, we discuss the large time behaviour of the solutions of two-dimensional stochastic convective Brinkman–Forchheimer (SCBF) equations on bounded domains. Under the functional setting , where and are appropriate separable Hilbert spaces and the fact that the embedding is compact, we establish the existence of random attractors in for the stochastic flow generated by 2D SCBF equations perturbed by additive noise. We prove the upper semicontinuity of the random attractors for 2D SCBF equations in , when the coefficient of random term approaches zero. Moreover, we obtain the existence of random attractors in a more regular space , using the pullback flattening property. The existence of random attractors ensures the existence of invariant compact random set and hence we show the existence of an invariant measure for 2D SCBF equations. Finally, we also comment on the uniqueness of invariant measures.

Asymptotic Behavior of the Kirchhoff Type Stochastic Plate Equation on Unbounded Domains

In this paper, we study the asymptotic behavior of solutions to the Kirchhoff type stochastic plate equation driven by additive noise defined on unbounded domains. We first prove the uniform estimates of solutions and then establish the existence and upper semicontinuity of random attractors.

Dynamics of stochastic nonlocal reaction–diffusion equations driven by multiplicative noise

This paper deals with fractional stochastic nonlocal partial differential equations driven by multiplicative noise. We first prove the existence and uniqueness of solution to this kind of equations with white noise by applying the Galerkin method. Then, the existence and uniqueness of tempered pullback random attractor for the equation are ensured in an appropriate Hilbert space. When the fractional nonlocal partial differential equations are driven by colored noise, which indeed are approximations of the previous ones, we show the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as [Formula: see text].

Existence and Upper Semi-Continuity of Random Attractors for Nonclassical Diffusion Equation with Multiplicative Noise on R<sup>n</sup>

This paper is concerned with the existence and upper semi-continuity of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity and multiplicative noise in H1(Rn). First, we study the existence and uniqueness of solutions by a noise arising in a continuous random dynamical system and the asymptotic compactness is established by using uniform tail estimate technique, and then the existence of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity. As a motivation of our results, we prove an existence and upper semi-continuity of random attractors with respect to the nonlinearity that enters the system together with the noise.

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

<p style='text-indent:20px;'>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.</p>

Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise

<p style='text-indent:20px;'>In this paper, we mainly consider the existence of random attractor and random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. First step, the well-posedness and the existence of a random attractor for the cocycle associated with the considered system is established. Second step, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero. Third step, we prove the regularity of random attractor in a higher regular space by the "iteration" method. Finally, we give the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor.</p>

Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise

<p style='text-indent:20px;'>In this paper, we study the asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise. The considered systems are driven by the fractional discrete Laplacian, which features the infinite-range interactions. We first prove the existence of pullback random attractor in <inline-formula><tex-math id="M1">\begin{document}$ \ell^2 $\end{document}</tex-math></inline-formula> for stochastic lattice systems. The upper semicontinuity of random attractors is also established when the intensity of noise approaches zero.</p>

Wong-Zakai approximations and pathwise dynamics of stochastic fractional lattice systems

<p style='text-indent:20px;'>This paper is concerned with the pathwise dynamics of stochastic fractional lattice systems driven by Wong-Zakai type approximation noises. The existence and uniqueness of pullback random attractor are established for the approximate system with a wide class of nonlinear diffusion term. For system with linear multiplicative noise and additive white noise, the upper semicontinuity of random attractors for the corresponding approximate system are also proved when the step size of the approximation approaches zero.</p>

Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift–Hohenberg equation with multiplicative noise

In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift–Hohenberg equations with multiplicative noise from two perspectives. First, by adopting the analytic semigroup theory, we prove the upper semi-continuity of random attractors in the Sobolev space H02(U), as the coefficient of the multiplicative noise approaches zero. Then, we extend the classical “stochastic Gronwall’s lemma,” making it more convenient in applications. Based on this improvement, we are allowed to use the analytic semigroup theory to establish the existence of ergodic invariant measures.

Wong–Zakai approximations and limiting dynamics of stochastic Ginzburg–Landau equations

This paper deals with the Wong–Zakai approximations and random attractors for stochastic Ginzburg–Landau equations with a white noise. We first prove the existence of a pullback random attractor for the approximate equation under much weaker conditions than the original stochastic equation. In addition, when the stochastic Ginzburg–Landau equation is driven by an additive white noise, we establish the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation tends to zero.

Large Time Behavior of Deterministic and Stochastic 3D Convective Brinkman-Forchheimer Equations in Periodic Domains

The large time behavior of deterministic and stochastic three dimensional convective Brinkman-Forchheimer (CBF) equations $$\begin{aligned} \partial _t{\varvec{u}}-\mu \Delta {\varvec{u}}+({\varvec{u}}\cdot \nabla ){\varvec{u}}+\alpha {\varvec{u}}+\beta |{\varvec{u}}|^{r-1}{\varvec{u}}+\nabla p={\varvec{f}},\ \nabla \cdot {\varvec{u}}=0, \end{aligned}$$ ∂ t u - μ Δ u + ( u · ∇ ) u + α u + β | u | r - 1 u + ∇ p = f , ∇ · u = 0 , for $$r\ge 3$$ r ≥ 3 ( $$\mu ,\beta >0$$ μ , β > 0 for $$r>3$$ r > 3 and $$2\beta \mu \ge 1$$ 2 β μ ≥ 1 for $$r=3$$ r = 3 ), in periodic domains is carried out in this work. Our first goal is to prove the existence of global attractors for the 3D deterministic CBF equations. Then, we show the existence of random attractors for the 3D stochastic CBF equations perturbed by small additive smooth noise. Furthermore, we establish the upper semicontinuity of random attractors for the 3D stochastic CBF equations (stability of attractors), when the coefficient of random perturbation approaches to zero. Finally, we address the existence and uniqueness of invariant measures of 3D stochastic CBF equations.

Wong–Zakai approximations of second-order stochastic lattice systems driven by additive white noise

In this paper, we study the Wong–Zakai approximations of a class of second-order stochastic lattice systems with additive noise. We first prove the existence of tempered pullback attractors for lattice systems driven by an approximation of the white noise. Then, we establish the upper semicontinuity of random attractors for the approximate system as the size of approximation approaches zero.

Wong–Zakai approximations and random attractors of non-autonomous stochastic discrete complex Ginzburg–Landau equations

In this paper, we consider the Wong–Zakai approximations and random attractors for stochastic discrete complex Ginzburg–Landau equations driven by nonlinear multiplicative noise. We establish the existence of the random attractor for the approximate system. For linear multiplicative noise or additive noise, we study the convergence of the solutions and the upper semicontinuity of random attractors for the approximate equation when the perturbation parameters tend to zero.

Wong–Zakai approximations and long term behavior of stochastic FitzHugh–Nagumo system

This paper is concerned with the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of stochastic FitzHugh–Nagumo system driven by white noise. We prove the existence and uniqueness of pullback random attractors for the approximate system under much weaker conditions than the original system. When the system is driven by additive white noise, we also prove the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.

Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems

In this paper, we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for the stochastic lattice systems. We first prove that the approximate system has a unique tempered pullback attractor under much weaker conditions than the original system. When the stochastic system is driven by a linear multiplicative noise or additive white noise, we prove the convergence of solutions and the upper semicontinuity of random attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero.