Topological dimensions of attractors for partial functional differential equations in Banach spaces

Wenjie Hu, Tomás Caraballo

Published 2023-03-07, updated 2023-04-23Version 2

The main object of this paper is to obtain estimations of Hausdorff as well as fractal dimensions of global attractors and pullback attractors for both autonomous and nonautonomous partial functional differential equations(PFDEs) in Banach spaces. New criterions for the finite Hausdorff and fractal dimensions of attractors in Banach spaces are firslty proposed by combing the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barriers caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of PFDEs based on the normal adjoint theory and obtain similar squeezing property. At last, the theoretical results are applied to three specific PFDEs, the retarded nonlinear reaction-diffusion equation, the retarded 2D-Navier-Stokes equation and the retarded semilinear wave equation. Optimal Hausdorff dimension estimation compared with previous works in Hilbert spaces and explicit bound on fractal dimensions that is not depend on the entropy number but only depend on the spectrum of the linearized equations are obtained.