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15 February 2025
 
  » arxiv » 1609.0179

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Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation
Greta Marino ; Sunra Mosconi ;
Date 1 Sep 2016
AbstractIn 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 $qle 0$ 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 $|u_q|_infty o 0$, as $qdownarrow 0$.
Source arXiv, 1609.0179
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