Random attractor for second-order stochastic delay lattice sine-Gordon equation

In this paper, we prove the existence of random $\mathcal{D}$ D -attractor for the second-order stochastic delay sine-Gordon equation on infinite lattice with certain dissipative conditions, and then establish the upper bound of Kolmogorov ε-entropy for the random $\mathcal{D}$ D -attractor.

Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with delays

<p style='text-indent:20px;'>We study asymptotically autonomous dynamics for non-autonom-ous stochastic 3D Brinkman-Forchheimer equations with general delays (containing variable delay and distributed delay). We first prove the existence of a pullback random attractor not only in the initial space but also in the regular space. We then prove that, under the topology of the regular space, the time-fibre of the pullback random attractor semi-converges to the random attractor of the autonomous stochastic equation as the time-parameter goes to minus infinity. The general delay force is assumed to be pointwise Lipschitz continuous only, which relaxes the uniform Lipschitz condition in the literature and includes more examples.</p>

Random attractors for stochastic delay wave equations on $ \mathbb{R}^n $ with linear memory and nonlinear damping

<p style='text-indent:20px;'>A non-autonomous stochastic delay wave equation with linear memory and nonlinear damping driven by additive white noise is considered on the unbounded domain <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. We establish the existence and uniqueness of a random attractor <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> that is compact in <inline-formula><tex-math id="M3">\begin{document}$ C{([-h, 0];H^1(\mathbb{R}^n))}\times C{([-h, 0];L^2(\mathbb{R}^n))}\times L_\mu^2(\mathbb{R}^+;H^1(\mathbb{R}^n)) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ 1\leqslant n \leqslant 3 $\end{document}</tex-math></inline-formula>.</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>

Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

<p style='text-indent:20px;'>This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional torus (<inline-formula><tex-math id="M2">\begin{document}$ n = 2, 3 $\end{document}</tex-math></inline-formula>):</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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}, \ \nabla\cdot\boldsymbol{u} = 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ r\geq1 $\end{document}</tex-math></inline-formula>. We prove that the global attractor of the above system is singleton under small forcing intensity (<inline-formula><tex-math id="M4">\begin{document}$ r\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ r\geq 3 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M9">\begin{document}$ r = n = 3 $\end{document}</tex-math></inline-formula>). But if one perturbs the above system with an additive or multiplicative white noise, there is no sufficient evidence that the random attractor keeps the singleton structure. We obtain that the random attractor for 2D stochastic CBF equations forced by additive and multiplicative white noise converges towards the deterministic singleton attractor for all <inline-formula><tex-math id="M10">\begin{document}$ 1\leq r&lt;\infty $\end{document}</tex-math></inline-formula>, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). For the case of 3D stochastic CBF equations perturbed by additive and multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for <inline-formula><tex-math id="M11">\begin{document}$ 3\leq r&lt;\infty $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M12">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ r = 3 $\end{document}</tex-math></inline-formula>), when the coefficient of random perturbation converges to zero.</p>

Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.

Fractal Dimension for the Nonautonomous Stochastic Fifth-Order Swift–Hohenberg Equation

Some dynamics behaviors for the nonautonomous stochastic fifth-order Swift–Hohenberg equation with additive white noise are considered. The existence of pullback random attractors for the nonautonomous stochastic fifth-order Swift–Hohenberg equation with some properties is mainly investigated on the bounded domain and unbounded domain, through the Ornstein–Uhlenbeck transformation and tail-term estimates. Furthermore, on the basis of some conditions, the finiteness of fractal dimension of random attractor is proved.