# Recurrent Solutions of a Nonautonomous Modified Swift-Hohenberg Equation

## TL;DR

This paper proves the existence of at least two recurrent solutions for a nonautonomous modified Swift-Hohenberg equation using Conley index theory, assuming the forcing term is recurrent.

## Contribution

It introduces a novel application of Conley index theory to establish multiple recurrent solutions in a nonautonomous PDE setting.

## Key findings

- Existence of at least two recurrent solutions under certain conditions.
- Application of Conley index theory to nonautonomous PDEs.
- Recurrent solutions depend on the recurrence of the forcing term.

## Abstract

We consider recurrent solutions of the nonautonomous modified Swift-Hohenberg equation $$u_t+\Delta^2u+2\Delta u+au+b|\nabla u|^2+u^3=g(t,x).$$ We employ Conley index theory to show that, if the forcing $g:\mathbb{R}\rightarrow L^2(\Omega)$ is a recurrent function, then there are at least two recurrent solutions in $H_0^2(\Omega)$ under appropriate assumptions on the parameters $a$, $b$ and $g$.