On $L^{\infty }$-estimates and the structure of the global attractor for weak solutions of reaction-diffusion equations

TL;DR

This paper investigates the structure of the global attractor for weak solutions of reaction-diffusion equations with non-regular nonlinearities, providing $L^{ abla}$-estimates and bounds on the attractor's dimension.

Contribution

It introduces $L^{ abla}$-estimates for weak solutions and refines the understanding of the attractor's structure and dimensional properties in reaction-diffusion systems.

Findings

Established $L^{ abla}$-estimates for weak solutions

Improved understanding of the attractor's structure

Derived bounds on Hausdorff and fractal dimensions

Abstract

In this paper, we study the structure of the global attractor for weak and regular solutions of a problem governed by a scalar semilinear reaction-diffusion equation with a non-regular nonlinearity, such that uniquness of solutions can fail to happen. First, using the Moser--Alikakos iterations we obtain the estimates of the weak solutions in the space $L^{\infty }(\Omega)$. After that, using these estimates we improve the existing results on the structure of the attractor. Finally, estimates of the Hausdorff and fractal dimension of the attractor are obtained.