Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation

TL;DR

This paper investigates the existence, non-existence, and asymptotic behavior of solutions to a generalized Swift-Hohenberg equation, establishing conditions for solutions' existence, non-existence, and decay as parameters vary.

Contribution

It provides new results on the existence of periodic solutions and conditions for solutions to vanish as parameters approach zero, expanding understanding of the equation's solution structure.

Findings

Non-existence of solutions for q ≤ 0

Existence of periodic solutions for small positive q

Solutions tend to zero as q approaches zero

Abstract

In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to $u""+qu"+f(u)=0$, under various hypotheses on the parameter $q$ and on the potential $F(t)=\int_0^tf(s)\, ds$, generally assumed to be bounded from below. We prove a non-existence result in the case $q\le 0$ and an existence result of periodic solution for: 1) almost every suitably small (depending on $F$), positive values of $q$; 2) all suitably large (depending on $F$) values of $q$. Finally, we describe some conditions on $F$ which ensure that some (or all) solutions $u_q$ to the equation satisfy $\|u_q\|_\infty\to 0$, as $q\downarrow 0$.