Topological dimensions of random attractors for stochastic partial differential equations with delay

Wenjie Hu; Tomás Caraballo

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arXiv · 2302.05501

Topological dimensions of random attractors for stochastic partial differential equations with delay

Wenjie Hu·Tomás Caraballo

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Abstract

The aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a class of stochastic partial differential equations with delay. The stochastic equation is first transformed into a delayed random partial differential equation by means of a random conjugation, which is then recast into an auxiliary Hilbert space. For the obtained equation, it is firstly proved that it generates a random dynamical system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation possesses random attractors by a uniform estimate of the solution and the asymptotic compactness of the generated RDS. After establishing the variational equation in the auxiliary Hilbert space and the $\mathbb{P}$ almost surely differentiable properties of the RDS, an upper estimate of both Hausdorff and fractal dimensions of the random attractors are obtained.

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Wenjie Hu, Tomás Caraballo. 2023-02-10. Topological dimensions of random attractors for stochastic partial differential equations with delay. https://arxiv.org/abs/2302.05501

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