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Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation. (English) Zbl 1411.34037
Authors’ abstract: In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to \[ u^{\prime \prime \prime \prime }+qu^{\prime \prime }+f(u)=0 \] under various hypotheses on the parameter \(q\) and on the potential \(F\left( t\right) =\int_{0}^{t}f\left( s\right) ds,\) generally assumed to be bounded from below. We prove a non-existence result in the case \(q\leq 0\) 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 \(\left\| u_{q}\right\| _{\infty }\rightarrow 0\) as \( q\downarrow 0\).

34B15 Nonlinear boundary value problems for ordinary differential equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
34C25 Periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: DOI
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