Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain

Wenjie Hu; Tomás Caraballo; Alain Miranville

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arXiv · 2311.09980

Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain

Wenjie Hu·Tomás Caraballo·Alain Miranville

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Abstract

The purpose of this paper is to investigate the existence and Hausdorff dimension as well as fractal dimension of global attractors for a delayed reaction-diffusion equation on an unbounded domain. The noncompactness of the domain causes the Laplace operator has a continuous spectrum, the semigroup generated by the linear part and the Sobolev embeddings are no longer compact, making the problem more difficult compared with the equations on bounded domains. We first obtain the existence of an absorbing set for the infinite dimensional dynamical system generated by the equation by a priori estimate of the solutions. Then, we show the asymptotic compactness of the solution semiflow by an uniform a priori estimates for far-field values of solutions together with the Arzel\`a-Ascoli theorem, which facilitates us to show the existence of global attractors. By decomposing the solution into three parts and establishing a squeezing property of each part, we obtain the explicit upper estimation of both Hausdorff and fractal dimension of the global attractors, which only depend on the inner characteristic of the equation, while not related to the entropy number compared with the existing literature.

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Wenjie Hu, Tomás Caraballo, Alain Miranville. 2023-11-16. Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain. https://arxiv.org/abs/2311.09980

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