Finite-Dimensionality of Tempered Random Uniform Attractors

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in $$L^2$$ L 2 is established by the squeezing approach and that in $$H_0^1$$ H 0 1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

Attractors for Multivalued Impulsive Systems: Existence and Applications to Reaction-Diffusion System

In this paper, we develop a general approach to investigate limit dynamics of infinite-dimensional dissipative impulsive systems whose initial conditions do not uniquely determine their long time behavior. Based on the notion of an uniform attractor, we show how to describe limit behavior of such complex systems with the help of properties of their components. More precisely, we prove the existence of the uniform attractor for an impulsive multivalued system in terms of properties of nonimpulsive semiflow and impulsive parameters. We also give an application of these abstract results to the impulsive reaction-diffusion system without uniqueness.

Uniform attractors for nonautonomous reaction-diffusion equations with the nonlinearity in a larger symbol space

<p style='text-indent:20px;'>Existence and structure of the uniform attractors for reaction-diffusion equations with the nonlinearity in a weaker topology space are considered. Firstly, a weaker symbol space is defined and an example is given as well, showing that the compactness can be easier obtained in this space. Then the existence of solutions with new symbols is presented. Finally, the existence and structure of the uniform attractor are obtained by proving the <inline-formula><tex-math id="M1">\begin{document}$ (L^{2}\times \Sigma, L^{2}) $\end{document}</tex-math></inline-formula>-continuity of the processes generated by solutions.</p>

Global existence, asymptotic behavior and uniform attractor for a non-autonomous Timoshenko system of type III with weak damping

In this paper, we investigate one-dimensional thermoelastic system of Timoshenko type III with double memory dampings. At first we give the global existence of solutions by using semigroup theory. Then we can prove the energy decay of solutions by constructing a series of Lyapunov functionals and obtain the existence of absorbing ball. Finally, we prove the asymptotic compactness by using uniform contractive functions and obtain the existence of uniform attractor.

Uniform attractors for a class of stochastic evolution equations with multiplicative fractional noise

This paper studies the (random) uniform attractor for a class of non-autonomous stochastic evolution equations driven by a time-periodic forcing and multiplicative fractional noise with Hurst parameter bigger than 1/2. We first establish the existence and uniqueness results for the solution to the considered equation and show that the solution generates a jointly continuous non-autonomous random dynamical system (NRDS). Moreover, we prove the existence of the uniform attractor for this NRDS through stopping time technique. Particularly, a compact uniformly absorbing set is constructed under a smallness condition imposed on the fractional noise.