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arXiv · 1609.00179

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

Greta Marino·Sunra Mosconi
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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 $q\downarrow 0$.

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Greta Marino, Sunra Mosconi. 2016-09-01. Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation. https://arxiv.org/abs/1609.00179

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