Recurrent Solutions of a Nonautonomous Modified Swift-Hohenberg Equation

Abstract

We consider recurrent solutions of the nonautonomous modified Swift-Hohenberg equation $$u_t+{\Delta }^{2}u+2\Delta u+au+b|\nabla u{|}^2+u^3=g(t,x).$$ We employ Conley index theory to show that, if the forcing $g:\mathbb{R}\to L^2(\Omega )$ 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 $g$.