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

• Published in: Nonlinearity Vol. 38; no. 7; pp. 75008 - 75035
• Main Authors: Hu, Wenjie, Caraballo, Tomás, Miranville, Alain
• Format: Journal Article
• Language: English
• Published: IOP Publishing 31.07.2025
• Subjects: 35B41; 35R10; 47H20; delay; fractal dimension; global attractors; Hausdorff dimension; Mathematics; reaction-diffusion equation; unbounded domain; Hausdorff dimension fractal dimension unbounded domain global attractors delay reaction-diffusion equation Mathematics Subject Classification numbers: 35R10 35B41 47H20
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/adde0b

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 to have 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 bounded domain case. We first obtain the existence of an absorbing set for the infinite dimensional dynamical system generated by the equation thanks to a priori estimates of the solution. Then, we show the asymptotic compactness of the solution semiflow by uniform a priori estimates for far-field values of solutions together with the Arzelà–Ascoli theorem, which facilitates us to show the existence of global attractors. By decomposing the solution into three parts and establishing squeezing properties of each part, we obtain the explicit upper bounds of both Hausdorff dimension and fractal dimension of the global attractors, which only depend on the inner characteristics of the equation, while not related to the entropy number compared with the existing literature.