Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential

TL;DR

This paper studies a reaction-diffusion system with obstacle constraints, proving the existence of finite-dimensional exponential and global attractors that describe the long-term behavior of solutions.

Contribution

It establishes the existence of finite-dimensional exponential and global attractors for reaction-diffusion problems with obstacle potentials in bounded domains.

Findings

Existence of exponential attractors with finite fractal dimension.

Global attractors are finite-dimensional.

Long-term solution behavior is characterized by these attractors.

Abstract

A reaction-diffusion problem with an obstacle potential is considered in a bounded domain of $\R^N$. Under the assumption that the obstacle $\K$ is a closed convex and bounded subset of $\mathbb{R}^n$ with smooth boundary or it is a closed $n$-dimensional simplex, we prove that the long-time behavior of the solution semigroup associated with this problem can be described in terms of an exponential attractor. In particular, the latter means that the fractal dimension of the associated global attractor is also finite.