Characterization of the attractor for nonautonomous reaction-diffusion equations with discontinuous nonlinearity

TL;DR

This paper investigates the long-term behavior of solutions to a reaction-diffusion equation with discontinuous nonlinearity, characterizing its attractor structure and relationships with various types of attractors.

Contribution

It provides a detailed analysis of the pullback attractor for a nonautonomous reaction-diffusion equation with discontinuous nonlinearity, including its structure and connections to other attractors.

Findings

The pullback attractor includes zero, a unique positive equilibrium, and heteroclinic connections.

Solutions exhibit specific regularity and uniqueness properties.

The attractor's relationship with other attractors is explicitly characterized.

Abstract

In this paper, we study the asymptotic behavior of the solutions of a nonautonomous differential inclusion modeling a reaction-diffusion equation with a discontinuous nonlinearity. We obtain first several properties concerning the uniqueness and regularity of non-negative solutions. Then we study the structure of the pullback attractor in the positive cone, showing that it consists of the zero solution, the unique positive nonautonomous equilibrium and the heteroclinic connections between them, which can be expressed in terms of the solutions of an associated linear problem. Finally, we analyze the relationship of the pullback attractor with the uniform, the cocycle and the skew product semiflow attractors.