\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 101, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/101\hfil Second-order Lagrangian systems]
{On the dynamics of second-order Lagrangian systems}

\author[R. Adams, W. D. Kalies, R. C. A. M. Vandervorst \hfil EJDE-2017/101\hfilneg]
{Ronald Adams, William D. Kalies, Robert C. A. M. Vandervorst}

\address{Ronald Adams \newline
Department of Mathematics,
Embry-Riddle Aeronautical University,
Daytona Beach, FL 32114, USA}
\email{adamsr25@erau.edu}

\address{William D. Kalies \newline
Department of Mathematics,
Florida Atlantic University,
Boca Raton, FL 33431, USA}
 \email{wkalies@fau.edu}

\address{Robert C. A. M. Vandervorst \newline
Department of Mathematics,
VU University,
Amsterdam, NL}
\email{vdvorst@few.vu.nl}

\dedicatory{Communicated by Paul Rabinowitz}

\thanks{Submitted October 6, 2016. Published April 11, 2017.}
\subjclass[2010]{37J45, 34C25, 47J30}
\keywords{Second-order Lagrangian; closed characteristic; twist system; 
\hfill\break\indent curve evolution}

\begin{abstract}
 In this article we are concerned with improving the twist condition for
 second-order Lagrangian systems. We characterize a local Twist property
 and demonstrate how results on the existence of simple closed
 characteristics can be extended in the case of the Swift-Hohenberg / extended
 Fisher-Kolmogorov Lagrangian. Finally, we describe explicit evolution
 equations for broken geodesic curves that could be used to investigate
 more general systems or closed characteristics.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}\label{sec:intro}

Second-order Lagrangian systems arise as fourth-order differential equations
obtained variationally as the  Euler-Lagrange equations of an action
functional which depends on the second derivative of the state
variable $u$ as well as its lower derivatives.  One important class of such
differential equations is
$u''''-\beta u'' +f(u)=0$, known as the Swift-Hohenberg
equation  for $\beta\leq0$ and the extended Fisher-Kolmogorov (eFK)
equation for $\beta>0$.
There have been numerous results concerning these equations,
see for instance \cite{AVV,GV,GBV,KKV2,KKV,KV,KV1,KW,PT,VV2,VV3}.

As is the case here, for the Hamiltonian system that is induced from
the second-order Lagrangian there is a natural two-dimensional section
for which bounded trajectories of the Hamiltonian system must intersect
in finitely or infinitely many distinct points.
We can characterize this section as $\Sigma_E=N_E\times\mathbb{R}$,
where $N_E$ is the one dimensional set
$$
N_E=\big\{(u,u''): \frac{\partial L}{\partial u''}u''-L(u,0,u'')=E\big\},
$$
and $H$ is the Hamiltonian and $E\in\mathbb{R}$.  The Hamiltonian flow
induces a return map to $\Sigma_E$ where closed trajectories correspond
to fixed points of the iterates of this map.
As in \cite{RV} the return map can often be viewed as an analogue of a
monotone area-preserving Twist map, Lagrangian systems that allow these
Twist maps are referred to as \textit{Twist systems}.

In this article we are concerned with improving the condition for a
second-order Lagrangian to possess a local version of the Twist condition,
 cf.\ assumption (A1) below. Through a version of the weak comparison
 principle we establish
that it is only necessary that the Twist condition
holds for a restricted class of laps which lie in a cone.
Exploiting the properties of this cone we can derive explicit formulas for
the vector field that describes the
dynamics of the endpoints between laps. The vector field can then be
estimated analytically or numerically.

The paper is organized as follows.
In Section~\ref{sec:prelim} we review the basic properties of second-order
Lagrangian systems and define the Twist property. We then characterize a
local Twist property and demonstrate how results on the existence of simple
closed characteristics can be extended in the case of
the SH/eFK Lagrangian.
In Section~\ref{sec:twist} we prove this extension and give another example
from the fifth-order KdV equation. In Section~\ref{sec:EE}, we describe
explicit evolution equations for  broken geodesic curves that
could be used to investigate other systems or non-simple closed characteristics,
and in Section~\ref{sec:example}
we briefly describe a framework for an analytic or numerical study.

\section{Preliminaries}\label{sec:prelim}

A second-order Lagrangian system is defined  by extremizing an action
functional of the form
\begin{equation*}
J[u]=\int_I L(u,u',u'')\,dt.
\end{equation*}
Computing the Euler-Lagrange equation yields
\begin{equation}\label{E-L}
\frac{d^2}{dt^2}\frac{\partial L}{\partial u''}
-\frac{d}{dt}\frac{\partial L}{\partial u'}+\frac{\partial L}{\partial u}=0.
\end{equation}
The Lagrangian action $J$ is invariant under the $\mathbb{R}$ action
$t\mapsto t+c$, which by Noether's Theorem yields the conservation law
\begin{equation}\label{Hamiltonian}
\Big(\frac{\partial L}{\partial u'}-\frac{d}{dt}\frac{\partial L}{\partial u''}
\Big)u'+\frac{\partial L}{\partial u''}u''-L(u,u',u'')=E.
\end{equation}

Under the natural hypothesis that $L$ is convex in $u''$, that is
 $\partial_w^2L(u,v,w)>0$ for all $(u,v,w)$,
the Lagrangian system $(L,dt)$ is equivalent to a Hamiltonian system on
$\mathbb{R}^4$ with the standard symplectic coordinates $x=(u,v,p_{u},p_v)$
endowed with the symplectic form $\omega$ given by
$\omega=du\wedge dp_{u}+dv\wedge dp_v$. The Hamiltonian is
\begin{align*}
H=p_{u}v+wL_w-L,
\end{align*}
where $w$ and $p_v$ are related by $p_v=L_w(u,v,w)$ and $p_{u}=L_v-p'_v$.
Stationary points of $J$ satisfy Equation~\eqref{Hamiltonian},
which is equivalent to $H(u,v,p_u,p_v)=E$. Thus, for the associated
Hamiltonian system $(H,\omega)$ the three-dimensional energy manifold
$M_E=\{(u,v,p_u,p_v)\in\mathbb{R}^4: H(u,v,p_u,p_v)=E\}$ is invariant.
  If $\nabla H\neq 0$ on $M_E$, then $E$ is a regular value, and $M_E$ is
a smooth non-compact manifold without boundary.
Equivalently, an energy level is regular if and only if $L_w(u,0,0)\neq0$
for all $u\in\mathbb{R}$ satisfying $L(u,0,0)+E=0$.

A bounded characteristic of a Lagrangian system $(L,dt)$ is a function
$u\in C^2_{b}(\mathbb{R},\mathbb{R})$ for which $\delta\int_{I}L(u,u',u'')=0$
with respect to variations $\delta u\in C^2_c(I,\mathbb{R})$ for any
compact interval $I\subset\mathbb{R}$.  Since the Lagrangian is a
$C^2$-function of $(u,v,w)$, it follows from the Euler-Lagrange equations that
$u\in C^3_b(\mathbb{R},\mathbb{R})$,
$L_w(\cdot)\in C^2_{b}(\mathbb{R},\mathbb{R})$, and
$(\frac{d}{dt}\frac{\partial L}{\partial w}-\frac{\partial L}{\partial v})(\cdot)
\in C^2_{b}(\mathbb{R})$.  Our main concern is to develop a new analytic
method to study the existence and structure of bounded, closed characteristics
on $M_E$, i.e.\ bounded characteristics which are periodic functions.
 We conclude this preliminary section with some relationships between closed
characteristics and the geometry of $M_E$.

Given an arbitrary $(2n-1)$-dimensional manifold $M$ embedded in
$(\mathbb{R}^{2n}, \omega)$, with $\omega$ being the standard symplectic form,
one can construct a Hamiltonian $H$ for which $M=H^{-1}(0)$.
The choice of $H$ is not instrinsic to finding periodic orbits,
it turns out that the geometry of $M$ is enough to describe them.
The geometry of $M$ and the symplectic $2$-form $\omega$ define a
characteristic line bundle,
\begin{align*}
\mathcal{E}_{M}=\{(x,\xi)\in T_{x}M\setminus\{0\}: w_{x}(\xi,\eta)=0\;
\forall\eta \in T_{x}M\}\subset TM\,.
\end{align*}
The vector field of any Hamiltonian $H$ with $M=H^{-1}(0)$ is a section of
$\mathcal{E}_{M}$.
 The trajectory of a periodic orbit can be viewed as a closed characteristic
of the line bundle, i.e.~an embedding $\gamma:S^{1}\to M$ of the
circle into $M$ for which
$$
T\gamma=\mathcal{E}_{M}\big|_{\gamma}.
$$
This gives us a relationship between existence of periodic solutions to
\eqref{E-L} and topological and geometric properties of its energy surfaces.
These closed characteristics can be found as critical points of the action
functional.  Weinstein \cite{W} conjectured in the 1970's, that any compact
hypersurface $M\in(\mathbb{R}^{2n},\omega)$, with the additional assumption that
$$
\alpha(\xi)\neq0\quad \text{for }0\neq\xi\in\mathcal{E}_{M},
$$
for some 1-form $\alpha$ with $d\alpha=\omega$, i.e.\ $M$ is of
\textit{contact type} relative to $\omega$, has at least one closed
characteristic.  This was proved later by Viterbo \cite{V}.
However this theory cannot be applied to energy manifolds determined by
second-order Lagrangian systems, because these manifolds are always non-compact
and they are not necessarily of contact type in $(\mathbb{R}^4,\omega)$,
as was proved in \cite{AVV}.

The existence of closed characteristics for second-order Lagrangian systems
has been studied  in several contexts, and we will summarize prior results
later in this section.
Following ideas first introduced in \cite{RV}, one method that has been used
to study closed characteristics variationally  is to divide
$u(t)$ into monotone segments.

\begin{definition} \rm
For $\ell<r$, an \textit{increasing lap} $u_{+}$ from $\ell$ to $r$ is a
solution to the Euler-Lagrange equation \eqref{E-L}
satisfying the boundary conditions $u(0)=\ell$, $u(A1)=r$, $u'(0)=u'(\tau)=0$
and $u'(t)>0$ for $0<t<\tau$ with a similar definition of a
\textit{decreasing lap} $u_{-}$.  A \textit{simple closed characteristic of type}
$(\ell,r)$ is a periodic solution which  is composed of a single increasing
lap from $\ell$ to $r$ followed by a single decreasing lap from $r$ to $\ell$
extended periodically.
\end{definition}

The concatenation of an increasing lap $u_+$ and a decreasing lap $u_-$  is
analogous to a broken geodesic and  is not necessarily a solution to \eqref{E-L}
 at the concatenation points, since the third derivatives need not agree there.
Setting $v=0$  in the Hamiltonian \eqref{Hamiltonian},   solutions satisfy
$wL_w-L=E$ at critical points.
Let $N$ denote this level set in the $(u,w)$-plane.  Then every simple
closed characteristic intersects $N$ exactly twice.  Moreover $N$ is a
section of $M$ given by $M\cap\{v=0\}$, and due to the convexity of $L$
in the $w$ variable, $N$ consists of two graphs in the $(u,w)$-plane.
That is, the projection $\pi$ of $N$ onto the $u$-axis can be described
by $\pi N=\{u: L(u,0,0)+E\geq 0\}$ and the sets $N\cap\{(u,w): u\geq 0\}$
and $N\cap\{(u,w): u\leq 0\}$ are graphs over $\pi N$.
 A particular connected component of $\pi N$ will be denoted by $I$, and
referred to as an \textit{interval component}.

Consider an interval component $I$, and define $B=\{(\ell,r)\in I^2:\ell<r\}$. For
simple closed characteristics one needs to find points $(\ell,r)\in B$ for which
the concatenation of an increasing and a decreasing lap is a solution.
For an increasing lap $u_{+}$ from $\ell$ to $r$ we let $p^{+}_{\ell}$ and
$p^{+}_{r}$ be the $p_{u}$-values at the concatenation points, and also
for a decreasing lap $u_{-}$ we let $p^{-}_{\ell}$ and $p^{-}_{r}$ be the
corresponding $p_u$ values.  If $u$ is the concatenation $u=u_{+}\#u_{-}$,
then necessary conditions for $u$ to be a solution of \eqref{E-L} are
$p^{+}_{\ell}=p^{-}_{\ell}$ and $p^{+}_{r}=p^{-}_{r}$ see \cite{RV}.
Since $u_{+}$ and $u_{-}$ are solutions to \eqref{E-L}, their intersection
 with $N$ determines the values of $u''=p_v$ uniquely from $\ell$ and $r$,
and we denote these values by $p_v(\ell)$ and $p_v(r)$.
Thus the necessary compatibility conditions on the $p_u$-values are also
sufficient.

\begin{figure}[ht] 
\begin{center}
\setlength{\unitlength}{1mm} 
\begin{picture}(102,48)(-7,0)
\qbezier(5,8)(12,8)(16,12)
\qbezier(16,12)(20,17)(24,24)
\qbezier(24,24)(28,31)(32,36)
\qbezier(32,36)(36,40)(40,40)
\qbezier(90,8)(83,8)(79,12)
\qbezier(79,12)(75,17)(71,24)
\qbezier(71,24)(67,31)(63,36)
\qbezier(63,36)(59,40)(55,40)
\put(2,0){\line(0,1){48}}
\put(-7,46){$u_{\rm max}$}
\put(35,43){$p_r^+$}
\put(58,43){$p_r^-$}
\put(-2,35){$r$}
\put(18,24){$u_+$}
\put(73,24){$u_-$}
\put(-2,9){$\ell$}
\put(5,3){$p_{\ell}^+$}
\put(86,3){$p_{\ell}^-$}
\put(-7,1){$u_{\rm min}$}
\end{picture}
\end{center}
\caption{The $p_v$ values at the endpoints of each lap are determined by
the minimum and maximum values $\ell$ and $r$, but the $p_{u}$
values are not, which gives a necessary and sufficient condition for
the concatenation $u_{-}\#u_{+}$ to be a simple closed characteristic,
$p_{r}^{+}=p_{r}^{-}$ and $p_{\ell}^{+}=p_{\ell}^{-}$.}
\label{lap}
\end{figure}

In \cite{RV} this broken geodesic method is used to prove the existence of
simple closed characteristics in systems where laps can be determined by
minimization. In particular fix an energy level $E$, and  let
\[
J_{E,\tau}[u]=\int_0^\tau L(u,u',u'')+E\,dt.
\]
Suppose that
\begin{itemize}
\item[(A1)]  for every pair $\ell\neq r$ there exists a unique minimizing lap
$u(t;\ell,r)$ of $J_E$
 defined on the interval $[0,\tau(\ell,r)]$, and both $u$ and $\tau$ are $C^1$
in $\ell$ and $r$.
\end{itemize}
This \emph{twist property} allows a reduction of the variational problem to
 finite dimensions by plugging the minimizer into the functional to obtain
the function $S_E(\ell,r)=J_E[u(t;\ell,r)]$.
\begin{align}\label{SE}
S_E(\ell,r)=\inf_{u\in X_{\tau},\tau\in\mathbb{R}^{+}}
\int^{\tau}_{0}L(u,u',u'')+E\,dt,
\end{align}
where $X_{\tau}=X_{\tau}(\ell,r)=\{u\in C^2([0,\tau]): u(0)=\ell,u(\tau)=r,
u'(0)=u'(\tau)=0, u'\big|_{(0,\tau)}>0$
if $\ell<r$,  and $u'\big|_{(0,\tau)}<0$ if $\ell>r \}$.
One then shows the existence of a critical point of $S_E(\ell,r)+S_E(r,\ell)$
in order to obtain  a simple closed characteristic.

The above results are extended in \cite{KV}  where a degree theoretic argument
is  used to establish the existence of a simple closed characteristic in systems
 which do not necessarily satisfy the twist condition via continuation to a
twist system. It is shown that for regular energy manifolds, the number of
closed characteristics can be bounded below by the second Betti number of $M_E$,
which in turn can be computed from the superlevel sets of the potential
function $L(u,0,0)+E\geq0$.  This result is
extended to singular energy levels in \cite{KW}.

Results concerning the existence of more general closed characteristics are
addressed  in \cite{GV, GBV}. For twist systems, a type of Conley-Morse theory
is used on the space of braids to obtain
forcing theorems for closed characteristics. It is shown that the set of closed
characteristics can have a rich structure. The degree arguments of \cite{KV}
do not extend these forcing results to systems
which do not necessarily satisfy the twist condition, because the degree for
many of these closed
characteristics is trivial even though the Conley index is nontrivial.

Our methods begin by reconsidering
\cite[Lemma 9]{RV}  which provides sufficient conditions for the twist property
to hold on an interval component for a Lagrangian of the form
$L(u,u',u'')=\frac{1}{2}|u''|^2+K(u,u')$.

\begin{lemma}[{\cite[Lemma 9]{RV}}] \label{lemma9_old}
Let $I_E$ be a connected component of $\pi^{u}N_E$.
Assume that
\begin{itemize}
\item[(a)] $\frac{\partial K}{\partial v}v-K(u,v)-E\leq 0$  for all
$u\in I_E$ and $v\in\mathbb{R}$;  and
\item[(b)] $\frac{\partial^2 K}{\partial v^2}v^2-\frac{5}{2}
\{\frac{\partial K}{\partial v}v-K(u,v)-E\}\geq 0$ for all
$u\in I_E$ and $v\in\mathbb{R}$.
\end{itemize}
Then for any pair $(\ell,r)\in I_E\times I_E\setminus\triangle$,
Problem \eqref{SE} has a unique minimizer
$(u,\tau)\in X_{\tau}\times\mathbb{R}^{+}$ (in fact the only critical point),
and the minimizer $u(t;\ell,r)$ depends $C^1$-smoothly on $(\ell,r)$
for $(\ell,r)\in \text{\em{int}$(I_E\times I_E\setminus\triangle)$}$.
\end{lemma}

Fix a regular energy level $E$.
We assume that $K(u,v)+E$ has the form
\begin{equation}\label{eq:K}
K(u,v)+E=\frac{\beta}{2}v^2+F(u),
\end{equation}
so that we consider Lagrangians of the eFK/Swift-Hohenberg type with
$$
L(u,u',u'')+E=\frac{1}{2}|u''|^2+\frac{\beta}{2}|u'|^2+F(u).
$$
From now on we will suppress the dependence on $E$.
The arguments we present apply to more general
form of the Lagrangian, as in Example~\ref{KdVexample} below, but we
consider this more restrictive case for
convenience and clarity of presentation.

In the next section we utilize a comparison principle to weaken
the sufficient conditions of Lemma~\ref{lemma9_old} and obtain the
following stronger result.

\begin{proposition}\label{prop:lem9}
For a Lagrangian of eFK/Swift-Hohenberg type,
let $I$ be a regular interval component and $I_1,I_2$ be subintervals of $I$.
There is a constant $C(\ell,r)>0$ such that if
\begin{align}\label{betaBD1}
\beta\leq\frac{2F((r-\ell)x+l)}{(r-\ell)C^{4/3}(\ell,r)\sin(\pi x)}\quad
\text{for all $x\in(0,1)$}
\end{align}
for all $(\ell,r)\in I_1\times I_2\setminus\triangle$ with $\ell<r$,
then the minimization problem in \eqref{SE}
has a local minimizer $(u,\tau)\in X_{\tau}\times\mathbb{R}^{+}$ for each
$(\ell,r)\in I_1\times I_2\setminus\triangle$. Moreover,
the family $u(t;\ell,r)$ depends $C^1$- smoothly on
$I_1\times I_2\setminus\triangle$.
\end{proposition}

 Note that the we need only consider $\ell<r$ in Proposition~\ref{prop:lem9}
because time-reversal of an increasing lap for $(\ell,r)$ yields a decreasing
lap for $(r,\ell)$. The proof of the
proposition is given in Section~\ref{sec:twist}.

Proposition~\ref{prop:lem9} allows the techniques in \cite{RV} to be applied
to a larger class of Lagrangians.  If Proposition~\ref{prop:lem9} is applied
over the whole interval component $I=[u_{\rm min},u_{\rm max}]$, i.e.\ on
$I\times I\setminus\triangle$, then one can obtain a stronger result than can
be attained using Lemma~\ref{lemma9_old}.
Indeed the hypotheses  of Lemma~\ref{lemma9_old}
 hold only for $\beta\le 0$.
However,
for $\ell=u_{\rm max}$ and $r=u_{\rm min}$
we have that
$$
\lim_{x\to 0^{+}}\frac{F((u_{\rm max}-u_{\rm min})x+u_{\rm min})}{\sin(\pi x)}
\quad\text{and}\quad
\lim_{x\to 1^{-}}\frac{F((u_{\rm max}-u_{\rm min})x+u_{\rm min})}{\sin(\pi x)}
$$
are strictly positive when $F$ has a simple zero at the endpoints of $I_E$,
which occurs for generic  potential functions $F(u)$, such as the double-well
potential and others, see  \cite[Sections 2.2.1, 2.2.2,  2.2.3]{RV}.

Therefore, in these cases
\begin{equation}\label{eqn:improved-twist-IC}
\min_{r,\ell\in I_E,\;x\in\Omega} \frac{F((r-\ell)x+\ell)}{\sin(\pi x)}>0,
\end{equation}
so that the local version of the twist property that is characterized in
Proposition~\ref{prop:lem9} holds for a range of positive values for
$\beta$ that can be estimated using \eqref{betaBD1}.
As an illustration we estimate the $\beta$ bound for a specific potential
function $F(u)$.

\begin{example}\label{example}\rm
Consider the double-well potential
\begin{align*}
F(u)=\frac{1}{4}(u^2-1)^2-\frac{1}{8}.
\end{align*}
A compact interval component is
$I=[-\sqrt{1-\frac{1}{\sqrt{2}}},\sqrt{1-\frac{1}{\sqrt{2}}}]$,
and it suffices to compute
\begin{align}\label{ex_betabd}
\inf_{[u_{\rm min},u_{\rm max}]}\inf_{\Omega}
\Big(\frac{\frac{1}{2}\left(((r-l)x+l)^2-1\right)^2
-\frac{1}{4}}{(r-\ell)C_2^{4/3}(\ell,r)\sin(\pi x)}\Big).
\end{align}
In Section \ref{subsec:mono} the following formula for the constant $C_2$
is given in equation~\eqref{in_lbd}
$$
C_2^{4/3}(\ell,r)=\sup_{\Omega}{\sqrt{\frac{8F((r-\ell)x+\ell)}
{3\pi^2\sin^2(\pi x)+\pi^2}}}
=\sup_{\Omega}{\sqrt{\frac{2\big(((r-l)x+l)^2-1\big)^2-1}
{3\pi^2\sin^2(\pi x)+\pi^2}}}.
$$
Using \eqref{ex_betabd}, we numerically estimate the upper bound on $\beta$ given
in equation~\eqref{betaBD1}  to be approximately $3.82$.
\end{example}

The existence of a simple closed characteristic of $L$ can then be proved
using \cite[Theorem 12]{RV}.
Note that simply the existence of a closed characteristic for all $\beta$
in the SH / eFK system follows
from the results in \cite{KV}. However, for $\beta>0$ this existence is
obtained via a degree argument,
and for the above range of positive $\beta$, we obtain existence variationally,
 which provides more information about the closed characteristics.

\begin{corollary}\label{cor:improved}
If condition~\eqref{eqn:improved-twist-IC} holds, then the hypotheses
of Proposition~\ref{prop:lem9} hold, and therefore there exists a simple
closed characteristic of $L$ on the interval component $I$.
\end{corollary}

In Section~\ref{sec:twist} we prove Proposition~\ref{prop:lem9}. Then we
define in Section~\ref{sec:EE} a type of curve evolution for which
closed characteristics are stationary points of the dynamics.
Finally, we briefly indicate how techniques from
dynamical systems might be applied using these methods in
Section~\ref{sec:example}.


\section{Localizing the twist property}\label{sec:twist}

Fix a regular  interval component
$I=[u_{\rm min},u_{\rm max}]$ with $F(u)>0$ on $(u_{\rm min},u_{\rm max})$,
and assume $u_{\rm min}, u_{\rm max}$
are simple zeros of $F$.
 Throughout this section $\ell$ and $r$ are arbitrary
points with $u_{\rm min}\le\ell<r\le u_{\rm max}$.

Next we introduce two changes of variables. Let
\[
z=v^{3/2}\quad\text{and}\quad x=\frac{u-\ell}{r-\ell}.
\]
The action functional $J:H^{1}_{0}(\Omega)\times I^2\to\mathbb{R}$
 written in the three variables $z,\ell,r$  is
\[
J[z,\ell,r]=\int_{\Omega}\frac{2}{9(r-\ell)}z_{x}^2
+\frac{(r-\ell)F((r-\ell)x+\ell)}{z^{2/3}}+\frac{(r-\ell)\beta}{2}z^{2/3}dx
\]
where $\Omega=(0,1)$. For fixed values of $\ell<r$,
the first change of variable ensures that the
gradient flow of $J$ in $H^1_0(\Omega)$ will be semilinear, and the second
change of coordinates provides the uniform spatial domain $[0,1]$.
Stationary functions of $J[z,\ell,r]$ satisfy the Euler -Lagrange equation
\begin{align*}
u''''-\frac{d}{dt}\frac{\partial K}{\partial u'}+\frac{\partial K}{\partial u}=0.
\end{align*}
Using the Hamiltonian relation, solutions of  this equation
satisfy
\begin{align*}
-u'u'''+\frac{1}{2}(u'')^2+\frac{\partial K}{\partial u'}u'-K(u,u')-E=0.
\end{align*}
Since $z$ is a function of $u$ on a lap, this can be expressed as
\begin{align}\label{Elleq_u}
z_{uu}=\frac{3}{2}\frac{\frac{\partial K}{\partial u'}u'-K(u,u')-E}{z^{5/3}},
\end{align}
and
 in terms of the variable $x$ we have
\begin{align}\label{ellipticEq}
 \frac{4}{9} z_{xx}=\frac{(r-\ell)^2}{3}
\Big(\frac{-2F((r-\ell)x+\ell)}{z^{5/3}}+\frac{\beta}{z^{1/3}}\Big).
\end{align}
for $K$ in \eqref{eq:K}.

\subsection{Monotonicity of the nonlinearity}\label{subsec:mono}

To take full advantage of the comparison principle for elliptic operators,
stated in Theorem~\ref{WCP} below, we need the following proposition.

\begin{proposition}\label{thm:mono}
If
\begin{equation}\label{eq:mono}
\beta\leq\frac{10F((\ell-r)x+l)}{z^{4/3}}\quad \text{for all $x\in\Omega$},
\end{equation}
then the nonlinearity
\[
N[z]=\frac{2}{3}\frac{F((\ell-r)x+\ell)}{z^{5/3}}-\frac{\beta}{3z^{1/3}}
\]
 is nonincreasing in $z$.
 \end{proposition}

\begin{proof}
The derivative with respect to $z$ is
\[
\frac{d}{dz}\Big(\frac{2}{3}\frac{F((\ell-r)x+l)}{z^{5/3}}
-\frac{\beta}{3z^{1/3}}\Big)
=\frac{-10}{9}z^{-8/3}F((\ell-r)x+l)+\frac{\beta}{9}z^{-4/3},
\]
which is nonpositive when condition \eqref{eq:mono} is satisfied.
\end{proof}


\subsection{Lower and upper solutions}\label{subsec:lus}

To consider the behavior of the solutions of \eqref{ellipticEq} near
$\partial\Omega$, we need to analyze the singular  operator
\[
E[z]=\frac{4}{9}z_{xx}+\frac{(r-\ell)^2}{3}
\Big(\frac{2F((r-\ell)x+\ell)}{z^{5/3}}-\frac{\beta}{z^{1/3}}\Big)
\]
for fixed $\ell<r$. The correct scaling between the $z_{xx}$ and $z^{-5/3}$
terms  is $z\sim x^{3/4}$, and hence we define the
conical shell
\[
\mathcal{C}=\big\{z\in L^{\infty}(\Omega):
C_1 d(x)^{3/4}\leq z(x)\leq C_2(d(x)^{3/4}+d(x))\;\forall x\in\Omega\big\}
\]
for some $C_1$, $C_2$ with $0<C_1<C_2$, where
$d(x)=\operatorname{dist}(x,\partial\Omega)$.

We use the following version of the weak comparison principle from Cuesta
and Tak\'{a}\v{c}  \cite{CuT} with $p=2$, see also Fleckinger-Pell\'{e}and
Tak\'{a}\v{c} \cite{JPT}.

\begin{theorem}[Weak comparison principle \cite{CuT}]\label{WCP}
Assume  $f\leq g$ in $L^{p/(p-1)}(\Omega)$, $\tilde{f}\leq\tilde{g}$
in $W^{1-(1/p),p}(\partial\Omega)$, and $u$, $v\in W^{1,p}(\Omega)$ are any
 weak solutions of the Dirichlet problems
\begin{gather*}
-\operatorname{div}(\mathbf{a}(x,\nabla u))-b(x,u)
=f(x)  \text{ in }\Omega; \quad
u=\tilde{f}\text{ on } \partial\Omega,\\
-\operatorname{div}(\mathbf{a}(x,\nabla v))-b(x,v)
=g(x)\text{ in }\Omega;\quad v=\tilde{g}\text{ on } \partial\Omega.
\end{gather*}
Then $u\leq v$ hold almost everywhere in $\Omega$ provided
$b(x,\cdot):\mathbb{R}\to\mathbb{R}$ is nonincreasing for a.e.\ $x\in\Omega$.
\end{theorem}

From Theorem \ref{WCP} we can now construct lower and upper solutions to
$E[z]=0$ following the arguments in Badra, Bal, and Giacomoni \cite{Ba}.
That is, we can find functions
$\underline{z},\bar{z}\in H^{1}_{0}(\Omega)\cap\mathcal{C}$ such that
$\underline{z}\leq\bar{z}$ and $E[\underline{z}]\ge 0$ and $E[\bar{z}]\le 0$
in $\Omega$.

Let $\phi_1$ denote the normalized positive eigenfunction associated with
the principal eigenvalue $\lambda_1$ of $-\Delta$ with homogeneous
Dirichlet boundary conditions. In this case $\phi_1(x)=\sin(\pi x)$ and
$\lambda_1=\pi^2$.  Since $\phi_1\in C^{1}(\overline{\Omega})$, we have
$\phi_1^{3/4}\in\mathcal{C}$.
We define the lower and upper solutions by
\[
\underline{z}=\eta\phi_1^{3/4}\quad\text{and}\quad
\overline{z}=M\phi_1^{3/4}
\]
for $\eta>0$ sufficiently small and $M>\eta$ sufficiently large.
For $z=C\phi_1^{3/4}$ with $C>0$
\begin{align*}
-E[z]=\frac{C\pi^2\phi_1^{3/4}}{3}+\frac{C}{12}\phi_1^{-5/4}(\partial_{x}\phi_1)^2
-\frac{2}{3}\frac{(r-\ell)^2F((r-\ell)x+\ell)}{{z}^{5/3}}
+\frac{(r-\ell)^2\beta}{3{z}^{1/3}}.
\end{align*}
Multiplying $-E[z]=0$ through by $12C^{5/3}\phi_1^{5/4}$ yields
\begin{align*}
C^{8/3}\left(4\pi^2\phi_1^2+(\partial_x\phi_1)^2\right)
+C^{4/3}4(r-\ell)^2\beta\phi_1-8(r-\ell)^2F((r-\ell)x+\ell)=0.
\end{align*}
Treating this as a quadratic in $C^{4/3}$ the nonnegative root is
\begin{equation} \label{eqn:roots}
\begin{aligned}
&\Big(-2\beta(r-\ell)^2\sin(\pi x) \\
&+ 2(r-\ell)\sqrt{(r-\ell)^2\beta^2\sin^2(\pi x)
+2F((r-\ell)x+\ell)\cdot\left(3\pi^2\sin^2(\pi x)+\pi^2\right)}\Big)\\
&\div \Big(3\pi^2\sin^2(\pi x)+\pi^2\Big).
\end{aligned}
\end{equation}
Therefore if we take  $C_1^{4/3}(\ell,r,\beta)\cdot(r-\ell)$ and
$C_2^{4/3}(\ell,r,\beta)\cdot(r-\ell)$ as lower and upper bounds for
the expression \eqref{eqn:roots} respectively, then for
$\eta=C_1(\ell,r,\beta)\cdot(r-\ell)^{3/4}$ and
$M=C_2(\ell,r,\beta)\cdot(r-\ell)^{3/4}$ we have that $\underline{z}$ and
$\bar{z}$ are lower and
upper solutions respectively, which leads us to a second cone
$$
\mathsf{C}(\ell,r,\beta)=\{z\in H^1_0(\Omega)\mid\underline{z}\leq z\leq\bar{z}\}.
$$
We can bound the function $z$ above independently of $\beta$ for $\beta>0$.
\begin{equation} \label{in_lbd}
\begin{aligned}
&\Big(-2\beta(r-\ell)^2\sin(\pi x) \\
&+ 2(r-\ell)\sqrt{(r-\ell)^2\beta^2\sin^2(\pi x)+2F((r-\ell)x+\ell)
\cdot\left(3\pi^2\sin^2(\pi x)+\pi^2\right)}\Big)\\
&\div \Big(3\pi^2\sin^2(\pi x)+\pi^2\Big) \\
&\leq \frac{(r-\ell)\sqrt{8F((r-\ell)x+\ell)\cdot
\left(3\pi^2\sin^2(\pi x)+\pi^2\right)}}
{3\pi^2\sin^2(\pi x)+\pi^2} \\
&=(r-\ell)\sqrt{\frac{8F((r-\ell)x+\ell)}{3\pi^2\sin^2(\pi x)+\pi^2}}
\end{aligned}
\end{equation}
We denote the maximum of $\sqrt{\frac{8F((r-\ell)x+\ell)}{3\pi^2\sin^2(\pi x)+\pi^2}}$
over $\Omega$ by $C_2^{4/3}(\ell,r)$.

 For $\beta>0$, restricting to the cone $\mathsf{C}(\ell,r,\beta)$ where
$z\le\bar{z}$, the  condition
\begin{equation} \label{betaBD}
\beta\leq\frac{10F((r-\ell)x+l)}{(r-\ell)C^{4/3}_2(\ell,r)\sin(\pi x)}\quad
\text{for all } x\in\Omega
\end{equation}
guarantees that  $N[z]$ is nonincreasing for $z\in\mathsf{C}(\ell,r,\beta)$.
Note that this inequality is
similar to \eqref{betaBD1} in Proposition~\ref{prop:lem9}.
Inequality \eqref{betaBD} is necessary, but
not sufficient for the existence of local minimizers, see the proof
 of Proposition~\ref{prop:lem9} below.

With this in mind, we compare \eqref{betaBD} with Lemma~\ref{lemma9_old},
which provides sufficient
conditions for the twist property to hold on an interval component $I$.
Hypothesis~(b) of Lemma~\ref{lemma9_old} is
\[
\frac{\partial^2 K}{\partial v^2}v^2
-\frac{5}{2}\big\{\frac{\partial K}{\partial v}v-K(u,v)-E\big\}\geq 0
\quad \text{for all $u\in I_E$ and $v\in\mathbb{R}$.}
\]
In our setting for eFK/Swift-Hohenberg type Lagrangians, this hypothesis reduces to
$$
v^2\beta\le10F(u)\quad \text{for all $u\in I$ and $v\in\mathbb{R}$},
$$
which holds only when $\beta\le 0$.
Note that when we consider functions $z(x)$ with $z=v^{3/2}$ and
$u=(r-\ell)x+\ell$, we obtain exactly  condition~\eqref{eq:mono}, so that
Hypothesis~(b) of Lemma~\ref{lemma9_old}
is sufficient, but not necessary, for the monotonicity of the operator $N[z]$.
 Therefore
while considering the cone $\mathsf{C}(\ell,r,\beta)$, it is natural to replace
condition Hypothesis (b) of Lemma~\ref{lemma9_old} with \eqref{betaBD},
which leads to the more general Proposition~\ref{prop:lem9}.
In particular, as shown in Example~\ref{example},
the inequality \eqref{betaBD} estimates the
range of positive $\beta$ values for which the twist property holds, whereas
Lemma~\ref{lemma9_old} requires $\beta\le 0$.
Note that the constant in \eqref{betaBD} is $10$ while the constant in
\eqref{betaBD1}  in Proposition~\ref{prop:lem9} is $2$. This discrepancy
is explained in the following proof.


\begin{proof}[Proof of Proposition \ref{prop:lem9}]
For existence and uniqueness we can apply the results of Crandall, Rabinowitz,
and Tartar \cite{CRT}, where the factor of 2 in \eqref{betaBD1}
 is necessary to ensure that
$\frac{2F((r-\ell)x+\ell)}{z^{5/3}}-\frac{\beta}{z^{1/3}}\to +\infty$ as
$z\to 0^{+}$ in Equation \eqref{ellipticEq}. The $C^1$ dependence on $\ell$
and $r$ follows from the same arguments as in the proof of \cite[Lemma 9]{RV}.
\end{proof}

\begin{example}\label{KdVexample} \rm
\textit{Fifth-order KdV equation}.  Here we briefly describe another example
of  how the results from this section extend the known results concerning
the existence of simple closed characteristics.
Consider the Lagrangian
\begin{equation} \label{lagrangian}
L(u,v,w)=\frac{1}{2}w^2+\frac{1}{2}\left(\alpha+2\mu u\right)v^2
+\frac{\kappa}{3}u^3+\frac{\sigma}{2}u^2
\end{equation}
which describes traveling waves in a fifth-order Korteweg-de Vries equation
(see \cite{PT})
$$
u_{t}+\gamma u_{xxxxx}+\beta u_{xxx}-\alpha\{2uu_{xx}+(u_x)^2\}_x
+2\kappa uu_x+3ru^2u_x=0.
$$
The techniques from the proof of Proposition~\ref{prop:lem9} can be adapted
to include the Lagrangian \eqref{lagrangian} and possibly more general
Lagrangians of the form $\mathcal{L}(u,v,w)=\frac{1}{2}w^2+K(u,v)$ of which $L$
is a special case, note that when $\mu=0$ $L$ reduces to the eFK/Swift-Hohenberg
Lagrangian.
Applying Lemma~\ref{lemma9_old} to $L$ gives the  inequality
\begin{equation} \label{KdV}
\alpha+2\mu u\leq 10\left(\frac{\kappa}{3}u+\frac{\sigma}{2}\right)
\frac{u^2}{v^2}\quad \text{for } u\in I_E, \; v\in\mathbb{R},
\end{equation}
which would require that $\alpha+2\mu u\leq 0$ to obtain the existence of
simple closed characteristics, cf.\ \cite[Section 2.2.3]{RV}.

Assume $\kappa$, $\mu>0$ and consider an interval component in
$\left(-\frac{3\sigma}{2\kappa},0\right)$, if we utilize Theorem~\ref{WCP}
and construct upper and lower solutions for equation~\eqref{Elleq_u},
then it is sufficient that inequality \eqref{KdV} hold for $v$ in a
cone defined by upper and lower solutions.  Thus no longer requiring
$\alpha+2\mu u\leq 0$, we would have an estimate of the form
\[
\alpha+2\mu \left((r-\ell)x+\ell\right)
\leq 10\left(\frac{\kappa}{3}\left((r-\ell)x+\ell\right)
 +\frac{\sigma}{2}\right)\frac{\left((r-\ell)x
 +\ell\right)^2}{(\underline{z}(x))^{4/3}}\quad \text{for } x\in \Omega,
\]
where  $I=[\ell,r]$, $u=(r-\ell)x+\ell$.  Note that using the explicit form
of the lower solution $\underline{z}$, the right hand side term can be bounded
away from zero, which gives the existence of simple closed characteristics
for a range of positive values for $\alpha+2\mu u$.
\end{example}

\section{Evolution equations}\label{sec:EE}

The twist condition, as discussed in the previous section, concerns the existence
of laps for fixed endpoints $\ell,r$, which are found as solutions to the
degenerate elliptic equation
\[
E[z]=\frac{4}{9}z_{xx}+\frac{(r-\ell)^2}{3}
\Big(\frac{2F((r-\ell)x+\ell)}{z^{5/3}}-\frac{\beta}{z^{1/3}}\Big)=0.
\]
In this section, we embed the problem of finding a closed characteristic
into a dynamical system where the endpoints $\ell,r$ evolve, which would
allow tools from dynamical systems theory, such as the Conley index, to be applied.
 There are several ways of constructing
such a dynamical system. We begin by considering the gradient dynamics
generated by the action functional
\begin{align}\label{actionJ}
J[z,\ell,r]=\int_{\Omega}\frac{2}{9(r-\ell)}z_{x}^2
+\frac{(r-\ell)F((r-\ell)x+\ell)}{z^{2/3}}+\frac{(r-\ell)\beta}{2}z^{2/3}\,dx.
\end{align}
The evolution equations arise by computing the total variation of
$J:H^{1}_{0}(\Omega)\times I^2\to\mathbb{R}$ as a function of three variables
 $z,\ell,r$.

To avoid the singularity at $z=0$ we consider the perturbed functional
\begin{equation}
J^{\epsilon}[z,\ell,r]=\int_{\Omega}\frac{2}{9(r-\ell)}z_{x}^2
+\frac{(r-\ell)F((r-\ell)x+\ell)}{(z+\epsilon)^{2/3}}
+\frac{(r-\ell)\beta}{2}(z+\epsilon)^{2/3}dx
\end{equation}
for $\epsilon>0$
so that
\begin{gather} \label{eqn:fullgradient}
(r-\ell)\delta_{z}J^{\epsilon}[z,\ell,r]
=-\frac{4}{9} z_{xx} - \frac{(r-\ell)^2}{3}\Big(\frac{2F((r-\ell)x+\ell)}
{\left(z+\epsilon\right)^{5/3}}+\frac{\beta}{\left(z+\epsilon\right)^{1/3}}
\Big)\cdot \delta z \\
\begin{aligned}
(r-\ell)^2\delta_{\ell}J^{\epsilon}[z,\ell,r] 
&=\frac{2}{9}\int_{\Omega}z_x^2 dx+(r-\ell)^2\int_{\Omega}\frac{(r-\ell)(1-x)F'-F}
 {(z+\epsilon)^{2/3}} dx \\
&\quad -(r-\ell)^2\int_\Omega\frac{\beta}{2}(z+\epsilon)^{2/3}
 dx\cdot\delta \ell,
\end{aligned}\\
\begin{aligned}
(r-\ell)^2\delta_{r}J^{\epsilon}[z,\ell,r] 
&=-\frac{2}{9}\int_{\Omega}z_x^2 dx+(r-\ell)^2\int_{\Omega}\frac{(r-\ell)x F'+F}
{(z+\epsilon)^{2/3}}\,dx  \\
&\quad +(r-\ell)^2\int_\Omega\frac{\beta}{2}(z+\epsilon)^{2/3}
dx\cdot\delta r.
\end{aligned}
\end{gather}
From this we obtain the  system
\begin{equation} \label{Ste}
\begin{gathered}
 z_t=\frac{4}{9} z_{xx} + \frac{(r-\ell)^2}{3}
\Big(\frac{2F((r-\ell)x+\ell)}{\left(z+\epsilon\right)^{5/3}}
-\frac{\beta}{\left(z+\epsilon\right)^{1/3}}\Big) ,\\
\begin{aligned}
\ell_t&=-\frac{2}{9}\int_{\Omega}z_x^2 dx
 -(r-\ell)^2\int_{\Omega}\frac{(r-\ell)(1-x)F'-F}{(z+\epsilon)^{2/3}} dx\\
&\quad  +(r-\ell)^2\int_\Omega\frac{\beta}{2}(z+\epsilon)^{2/3} dx,
\end{aligned}\\
\begin{aligned}
r_t&=\frac{2}{9}\int_{\Omega}z_x^2 dx
 -(r-\ell)^2\int_{\Omega}\frac{(r-\ell)x F'+F}{(z+\epsilon)^{2/3}}\,dx \\
&\quad -(r-\ell)^2\int_\Omega\frac{\beta}{2}(z+\epsilon)^{2/3} dx.
\end{aligned}
  \end{gathered}
\end{equation}
where the argument of $F$ and $F'$ is $(r-\ell)x+\ell$. Let
$Q_{T}=\Omega\times(0,T)$  and $\Sigma_{T}=\partial\Omega\times(0,T)$.
  System \eqref{Ste}  allows the possibility of studying laps and closed
characteristics as stationary points of a type of curve evolution in the plane.
 Indeed the evolution can be cast as a curve shortening type flow under a
 Finsler form induced by the Lagrangian \cite{RA}.
 However, when $\epsilon=0$, this system is degenerate when $z=0$.

\begin{theorem}\label{thm:exist}
Given $(z_0,\ell_0,r_0)\in H^{1}_{0}(\Omega)\times I\times I$ with
$r_0\neq \ell_0$ and $z_0>0$ on $\Omega$ there exists
$T(z_0,\ell_0,r_0,\epsilon)>0$ such that a unique solution
$(z_\epsilon,\ell_\epsilon,r_\epsilon)$ in the space
$C^1([0,T];C^{\infty}(\Omega))\times C^1([0,T];I)\times C^1([0,T];I)$ to 
\eqref{Ste} exists, and
$J^{\epsilon}[z_\epsilon(t),\ell_\epsilon(t),r_\epsilon(t)]$
is decreasing on $[0,T]$.
\end{theorem}

\begin{proof}
Existence and uniqueness follow from standard semigroup theory, since
the perturbed nonlinearity is Lipschitz for $z+\epsilon>0$, see for
instance \cite{Henry}.  Moreover, $J^{\epsilon}$ decreases along
$(z_\epsilon,\ell_\epsilon,r_\epsilon)$ by construction
\[
\frac{d}{dt}J^{\epsilon}[z,\ell,r]
=-(r-\ell)(\nabla_{z}J^{\epsilon})^2-(r-\ell)^2(\nabla_{r}J^{\epsilon})^2
-(r-\ell)^2(\nabla_{\ell}J^{\epsilon})^2<0.
\]
\end{proof}

While it may be possible to analyze the dynamics of \eqref{Ste} directly,
for example by scaling the system so that the $z$-dynamics is fast compared to
the dynamics of $\ell, r$ and use geometric singular perturbation theory,
we take a different approach. We consider the finite-dimensional system obtained
by removing the parabolic PDE and substituting a family of stationary
solutions $z_{\epsilon,\infty}$ to the PDE into the evolution equations
for $\ell, r$ to obtain
\begin{equation} \label{eqn:reduced}
\begin{gathered}
\begin{aligned}
\ell_t&=-\frac{2}{9}\int_{\Omega}(\partial_xz_{\epsilon,\infty})^2 dx
 -(r-\ell)^2\int_{\Omega}\frac{(r-\ell)(1-x)F'-F}{(z_{\epsilon,\infty}
 +\epsilon)^{2/3}} dx \\
&\quad +(r-\ell)^2\int_\Omega\frac{\beta}{2}(z_{\epsilon,\infty}+\epsilon)^{2/3} dx,
\end{aligned}\\
\begin{aligned}
  r_t&=\frac{2}{9}\int_{\Omega}(\partial_xz_{\epsilon,\infty})^2 dx
 -(r-\ell)^2\int_{\Omega}\frac{(r-\ell)x F'+F}{(z_{\epsilon,\infty}
 +\epsilon)^{2/3}}\,dx \\
&\quad -(r-\ell)^2\int_\Omega\frac{\beta}{2}(z_{\epsilon,\infty}+\epsilon)^{2/3} dx.
\end{aligned}
  \end{gathered}
\end{equation}
Stationary solutions to this two-dimensional system are necessarily stationary
solutions to system \eqref{Ste}.

 To see that a smooth family of solutions $z_{\epsilon,\infty}$ exists, define
$
E_\epsilon[z]=E[z+\epsilon],
$
and recall the lower and upper solutions $\underline{z},\bar{z}$ of $E[z]$.
Then $\underline{z}-{\epsilon}$ is a lower solution for the regularized system since
 $$
E_\epsilon[\underline{z}-\epsilon]= \frac{4}{9}(\underline{z}-{\epsilon})_{xx}
+\frac{(r-\ell)^2}{3}\left(\frac{2F((r-\ell)x+\ell)}{(\underline{z}-{\epsilon}
+\epsilon)^{5/3}}-\frac{\beta}{(\underline{z}-{\epsilon}+\epsilon)^{1/3}}\right)
=E[\underline{z}]\ge 0.
 $$
For $\beta$ satisfying \eqref{betaBD},
$$
E_\epsilon[\bar{z}]= \frac{4}{9}\bar{z}_{xx}+N[\bar{z}+\epsilon]
\le \frac{4}{9}\bar{z}_{xx}+N[\bar{z}]
= E[\bar{z}]\le 0,
$$
and hence $\bar{z}$ is an upper solution for all $\epsilon\ge 0$.
 With this in mind we can define a cone for the $\epsilon$-perturbed flow
 $$
 \mathsf{C}(\ell,r,\beta,\epsilon)=\{z\in H^1_0(\Omega):
\underline{z}-{\epsilon}\leq z\leq\bar{z}\}.
 $$
Then  the methods used in \cite{Ba}  and \cite{CRT} imply that such a family exists
within the cones $\mathsf{C}(\ell,r,\beta,\epsilon)$, see Theorem~0.8 in \cite{Ba}.

These methods also establish the existence of time-dependent solutions for
the family of parabolic PDE's $z_t=E_\epsilon[z]$
for $\epsilon\ge 0$,  however, not necessarily for the coupled PDE-ODE
system \eqref{Ste} when $\epsilon=0$, since for $\epsilon=0$ the PDE is singular.

Here we state the existence and regularity for the following singular parabolic
equation where $r$, $\ell$ are fixed.
\begin{equation} \label{Pt}
z_t=\frac{4}{9} z_{xx} + \frac{(r-\ell)^2}{3}\Big(\frac{2F((r-\ell)x+\ell)}
{ z^{5/3}}-\frac{\beta}{ z^{1/3}}\Big).
\end{equation}

\begin{definition} \rm
Let
\[
\mathbf{V}(Q_T)=\{z:z\in L^{\infty}(Q_T),z_t\in L^2(Q_T),
z\in L^{\infty}(0,T;H^1_{0} (\Omega))\}.
\]
A \emph{weak solution} to \eqref{Pt} is a function $z\in \mathbf{V}(Q_T)$,
satisfying
\begin{enumerate}
 \item $\operatorname{ess\,inf}_{K}z>0$ for every compact $K\subset Q_{T}$;
 \item for every test function $\phi\in \mathbf{V}(Q_T)$,
 \[
 \int_{Q_{T}}\Big(\phi \frac{\partial z}{\partial t}
 +\frac{4}{9}z_{x}\phi_{x}-\frac{(r-l)^2}{3}\Big(\frac{2F((r-l)x+l)\phi}{z^{5/3}}
 -\frac{\beta\phi}{z^{1/3}}\Big)\Big) \,dx\,dt=0;
\]
 \item $z(0,x)=z_{0}(x)$ a.e. in $\Omega$.
\end{enumerate}
\end{definition}

\begin{remark}\rm
If $1/z^{5/3}\in L^{\infty}(0,T;H^{-1}(\Omega))$ then the integral in $(2)$
is well-defined.
\end{remark}

\begin{theorem}\label{reg}
 For $z_0\in H^1_0(\Omega)\cap \mathcal{C}$ and $\beta$ satisfying \eqref{betaBD}
 there  exists a unique weak solution $z$ to the parabolic equation \eqref{Pt}
which satisfies
 $z(t)\in C((0,\infty);H_0^{\frac{5}{4}-\frac{5}{2}\eta}(\Omega))$
for all $0<\eta<3/8$.
\end{theorem}

To obtain this regularity result we utilize the methods from
 \cite[Theorem 4.2]{Ba}, which require the interpolation theory of Sobolev spaces
(cf. Triebel \cite{Triebel}), and the Hardy Inequality.
Recall that the operator $A=-\frac{4}{9}\frac{d^2}{dx^2}$ has domain
$\mathcal{D}(A)=H^{1}_{0}(\Omega)\cap H^2(\Omega)$, hence for fractional
powers of $A$, $A^{\theta}$ with domain $\mathcal{D}(A^{\theta})$ in
$L^2(\Omega)$ we have the following fact.

\begin{proposition}\label{sobolev}\quad
\begin{itemize}
\item [(i)] $\mathcal{D}(A^{\theta})=(\mathcal{D}(-A),L^2(\Omega))_{1-\theta,2}$;
\item[(ii)] $\mathcal{D}(A^{\theta})=H^{2\theta}_{0}$ if
$1/4<\theta< 1$;
\item[(iii)] $A^{\theta}$ is an isomorphism from $\mathcal{D}(A)$ onto
$\mathcal{D}(A^{1-\theta})$ as well as from $L^2(\Omega)$ onto the dual space
$(\mathcal{D}(A^{\theta}))'$.
\end{itemize}
\end{proposition}
See \cite[Proposition 4.1]{Ba}, also one can look in \cite{Triebel}.

\begin{lemma}\label{dual}
Let $\theta\in[0,1)$ and $q>\frac{2}{1-\theta}$.  For $0\leq t_0<T<\infty$
let $L_{T}$ be the linear operator defined by $L_{T}(f)=u$ where $u$
is the solution to
\begin{equation}
  \begin{gathered}
    u_t-\frac{4}{9}u_{xx}=f \quad\text{in } Q_{T},\\
    u=0 \quad \text{on }\Sigma_{T}, \\
  u(0)=0 \quad  \text{in } \Omega.
  \end{gathered}
\end{equation}
Then $L_{T}$ is a bounded operator from $L^{q}(t_0,T;(\mathcal{D}(A^{\theta}))')$
into $\mathcal{X}_{q,\theta,T}$ as well as from
$L^{q}(t_0,T;(\mathcal{D}(A^{\theta}))')$ into
$C([t_0,T],\mathcal{D}(A^{1-\theta-\frac{2}{q}}))$.
\end{lemma}

For a proof of the above lemma, see \cite[Lemma 4.4]{Ba}.
In particular we have the  inequality
\begin{equation} \label{Reg:ineq}
\|L_{T}f\|_{C([t_0,T],\mathcal{D}(A^{1-\theta-\frac{2}{q}}))}
\leq C\|f\|_{L^q(t_0,T;(\mathcal{D}(A^{\theta}))')}.
\end{equation}

We make use of the following Hardy type inequality, see \cite[Lemma 4.5]{Ba} or
\cite[Lemma 3.2.6.1]{Triebel}.

\begin{lemma}\label{Hardy}
Let $s\in[0,2]$ such that $s\neq 1/2$ and $s\neq 3/2$.
Then the following generalization of Hardy's inequality holds:
\begin{equation} \label{hardy_ineq}
\|d^{-s}g\|_{L^2(\Omega)}\leq C\|g\|_{H^s(\Omega)}\quad \text{for all }
g\in H^s_0(\Omega).
\end{equation}
\end{lemma}

\begin{proof}[Proof of Theorem~\ref{reg}]
The existence and uniqueness of weak solutions to \eqref{Pt} follows
from a straightforward modification of the techniques in \cite{Ba}
to accommodate singular terms of the
form $F(x)z^{-5/3}$ with $F(x)$ bounded and positive a.e.,
instead of simply $z^{-5/3}$, and the addition of a weaker singular
term $\beta z^{-1/3}$. In particular see
\cite[Theorem 0.16]{Ba}.

To obtain regularity we proceed as follows.
For $t>0$ choose $t_0,T>0$ such that $t_0<t<T$.
 Note that the solution $z$ to $P_t$ on $(0,T)$ satisfies
\[
z(t)=e^{-At}z_0+L_{T}\Big(\frac{(r-l))^2}{3}\Big(\frac{2F}{z^{5/3}}
-\frac{\beta}{z^{1/3}}\Big)\Big).
\]
Since $0<\eta<\frac{3}{8}$, we have $\theta=\frac{5}{4}-\frac{5}{2}\eta>\frac{1}{4}$
as in Proposition \ref{sobolev}, and since $z\in\mathcal{C}$, we know that
$\frac{1}{z^{5/3}}=O\big(\frac{1}{d(x)^{5/4}}\big)$.
Therefore by Lemma \ref{Hardy} we have
$$
\frac{(r-l))^2}{3}\Big(\frac{2F}{z^{5/3}}
-\frac{\beta}{z^{1/3}}\Big)\in C( (0,T],(H^{\frac{3}{4}+\frac{\eta}{2}}_0(\Omega))').
$$
Indeed setting $s=\frac{3}{4}+\frac{\eta}{2}$
in \eqref{hardy_ineq}, for $g\in H^{\frac{3}{4}+\frac{\eta}{2}}_0$ we obtain
\begin{align*}
\|\frac{1}{z^{5/3}}g\|^2_{L^2(\Omega)}
&\leq\int_{\Omega}\Big|\frac{g}{C_1 d(x)^{5/4}}\big|^2dx \\
&=\frac{1}{C_1^2}\int_{\Omega}\Big|\frac{d(x)^{3/4+\eta/2}}{d(x)^{5/4}}
\frac{g}{d(x)^{3/4+\eta/2}}\Big|^2dx\\
&\leq\frac{1}{C_1^2}\|d^{-1/2+\eta/2}\|_{L^2(\Omega)}\|d(x)^{-(3/4+\eta/2)}
g\|_{L^2(\Omega)} \\
&\leq\frac{1}{C_1^2}\|d^{-1/2+\eta/2}\|_{L^2(\Omega)}\|g\|_{H_0^{3/4+\eta/2}}.
\end{align*}
Furthermore, $C((0,T],(H^{\frac{3}{4}+\frac{\eta}{2}}_0(\Omega))')
=C((0,T],(\mathcal{D}(A)^{\frac{3}{8}+\frac{\eta}{4}})')$
from Proposition~\ref{sobolev}.  Applying Lemma~\ref{dual} with
$q=\frac{2}{\eta}$ and $\theta=\frac{3}{8}+\frac{\eta}{4}$ we see that
$$
L_{T}\Big(\frac{(r-l))^2}{3}\Big(\frac{2F}{z^{5/3}}-\frac{\beta}{z^{1/3}}\Big)\Big)
\in C([0,T],D(A^{1-\theta-\frac{2}{q}}))=C([0,T],D(A^{\frac{5}{8}-\frac{5}{4}\eta})).
$$
Therefore,
$$
L_{T}\Big(\frac{(r-l))^2}{3}\Big(\frac{2F}{z^{5/3}}-\frac{\beta}{z^{1/3}}\Big)
\Big)\in C([0,T],H_0^{\frac{5}{4}-\frac{5}{2}\eta}(\Omega)),
$$
and
\[
t\to z(t)=e^{-At}z_0+L_{T}\Big(\frac{(r-l)^2}{3}
\Big(\frac{2F}{z^{5/3}}-\frac{\beta}{z^{1/3}}\Big)\Big)
\in C((0,T],H_0^{\frac{5}{4}-\frac{5}{2}\eta}(\Omega)).
\]
Since $t$ is arbitrary,
$z(t)\in C((0,\infty),H_0^{\frac{5}{4}-\frac{5}{2}\eta}(\Omega))$.
\end{proof}

We now obtain the following existence and stabilization result.

\begin{theorem}\label{stable}
Fix $u_{\rm min}<\ell<r<u_{\rm max}$.
If $\beta$ satisfies the bound in \eqref{betaBD}, then there exists a unique
$z_{\epsilon,\infty}\in H^1_0(\Omega)\cap\mathsf{C}(\ell,r,\beta,\epsilon)
\cap C_0(\overline{\Omega})$ satisfying
\[
E_\epsilon[z_{\epsilon,\infty}]=0.
\]
Moreover, for each $z_0\in\mathsf{C}(\ell,r,\beta,\epsilon)$ there exists 
a unique solution $z(t)$ of
\begin{equation} \label{parbpde}
     z_t=E_\epsilon[z] \quad \text{for $t\in(0,\infty)$ with $z(0)=z_0$,}
\end{equation}
and $z(t)\to z_{\epsilon,\infty}$ as $t\to\infty$ in $H^1_0(\Omega)$.
Furthermore, there exists $M(\ell,r,\beta)>0$
such that
\[
\|z_{\epsilon,\infty}\|_{H_0^{9/8}(\Omega)}\le M(\ell,r,\beta) \quad
\text{for $\epsilon\ge 0$.}
\]
\end{theorem}

\begin{proof}
First, as described in the proof of Theorem~\ref{reg},  we note that
the techniques in \cite{Ba} can be modified
to apply to the type of nonlinearity we are considering here.
In particular Theorem~0.8 in \cite{Ba} establishes the existence and uniqueness
of the function $z_\infty\in H^1_0\cap\mathcal{C}$ with $E[z_\infty]=0$, and
\cite[Theorem 0.15]{Ba} can then be applied to
obtain convergence  of $z(t)\to z_{\infty}$ in $L^{\infty}(\Omega)$.

Now setting $\eta=1/20$ in Theorem \ref{reg} yields 
$z(t)\in C((0,\infty),H_0^{9/8}(\Omega))$.
Fix $\tau>0$ and for each $t>0$ choose $0<t_0<t<t_0+\tau=T$. Then
$$
t\to z(t)= e^{-tA}z_0+L_{T}\Big(\frac{(r-\ell)^2}{3}
\big(\frac{2F}{z^{5/3}}-\frac{\beta}{z^{1/3}}\big)\Big).
$$
Since $\underline{z}\leq z(t)\leq \bar{z}$, we can apply Lemma~\ref{dual}
\begin{equation}\label{eqn:finalbound}
\begin{aligned}
 &\|L_{T}\Big(\frac{(r-\ell)^2}{3}\big(\frac{2F}{z^{5/3}}
 -\frac{\beta}{z^{1/3}}\big)\Big)\|_{C([t_0,T],H_0^{9/8}(\Omega))}\\
 &\leq C\|\frac{(r-\ell)^2}{3}\big(\frac{2F}{z^{5/3}}-\frac{\beta}{z^{1/3}}
 \big)\|_{L^q(t_0,T;(H^{31/40}_0(\Omega))')}\\
&\leq C\|\frac{(r-\ell)^2}{3}\frac{2F}{z^{5/3}}\|_{L^q
 (t_0,T;(H^{31/40}_0(\Omega))')} \\
&\quad +C\|\frac{(r-\ell)^2}{3}\frac{|\beta|}{z^{1/3}}
 \|_{L^q(t_0,T;(H^{31/40}_0(\Omega))')}\\
&\leq C(\ell,r)\tau^{1/q}\|\frac{(r-\ell)^2}{3}
 \frac{2F}{z^{5/3}}\|_{L^q(t_0,T;(H^{31/40}_0(\Omega))')}\\
&\quad +C(\ell,r)\tau^{1/q}\|\frac{(r-\ell)^2}{3}
 \frac{|\beta|}{z^{1/3}}\|_{(H^{13/32}_0(\Omega))'}<\infty
\end{aligned}
\end{equation}

Using the Hardy inequality as in the proof of Theorem \ref{reg} we see that 
the last terms are bounded independent of $T$, and hence  
$\sup_{[1,\infty]}\|z(t)\|_{H_0^{9/8}(\Omega)}<\infty$. 
From the compactness of the embedding of $H_0^{9/8}$ into $H_0^1$
it follows that for every $z_0\in H_0^1\cap\mathcal{C}$ we have
$z(t)\to z_\infty$ as $t\to\infty$ in $H^1_0(\Omega)$.

For $\epsilon>0$ we can repeat the above arguments for the perturbed 
parabolic equation
\begin{equation} \label{Ptep}
z_t=\frac{4}{9} z_{xx} + \frac{(r-\ell)^2}{3}\Big(\frac{2F((r-\ell)x+\ell)}
{ (z+\epsilon)^{5/3}}-\frac{\beta}{ (z+\epsilon)^{1/3}}\Big)
\end{equation}
as follows.
Just as for the singular case $\epsilon=0$ addressed in Theorem~\ref{reg},
the techniques of \cite{Ba} can be suitably modified to establish the existence,
 uniqueness, and regularity to the parabolic PDE \eqref{Ptep}
for $\epsilon>0$. Indeed, standard semigroup theory
\cite{Henry} also applies in this case since the nonlinearity is not singular.
Recall that a lower solution for \eqref{Ptep} is given by $
\underline{z}_{\epsilon}=\underline{z}-\epsilon$, and the upper solution
$\bar{z}$ for $P_t$ is also an upper solution for \eqref{Ptep}.
Therefore the arguments used in the proof of Theorem \ref{reg} can be applied
to establish that $z_\epsilon(t)\in C((0,\infty),H_0^{9/8}(\Omega))$.
Note that since $z_\epsilon(t)>-\epsilon$, the solution exists for all $t\ge 0$.
Also recall the existence of a unique solution
$z_{\epsilon,\infty}\in H^1_0\cap\mathcal{C}$
to $E_\epsilon[z_{\epsilon,\infty}]=0$
with $z_\epsilon(t)\to z_{\epsilon,\infty}$ in
$L^\infty(\Omega)$ by Theorem~0.15 in \cite{Ba}.

Moreover since $\underline{z}_{\epsilon}\leq z_\epsilon(t)$, we have
$$
\frac{1}{\left(z_\epsilon(t)+\epsilon\right)^{p}}
\leq\frac{1}{\left(\underline{z}_{\epsilon}+\epsilon\right)^{p}}
=\frac{1}{\underline{z}^{p}} \quad \text{ for $p=5/3$ or $1/3$.}
$$
Hence the bound in equation \eqref{eqn:finalbound} holds for the solutions to
\eqref{Ptep} as well so that
$\sup_{[1,\infty]}\|z_\epsilon(t)\|_{H_0^{9/8}(\Omega)}<\infty$, and
for every $z_0\in H_0^1\cap\mathcal{C}$ we have  
$z_\epsilon(t)\to z_{\epsilon,\infty}$ in $H^1_0(\Omega)$.

Finally, taking $z_0=z_{\epsilon,\infty}$ so that 
$z_\epsilon(t)=z_{\epsilon,\infty}$ for all $t\ge0$, and using the fact that 
$e^{-tA}z_0\to 0$ in $H_0^{9/8}(\Omega)$, for each
$\epsilon>0$ there exists a time $t_1>0$ such that 
$\|e^{-tA}z_{\epsilon,\infty}\|_{H_0^{9/8}}\le 1$. Since the bound 
\eqref{eqn:finalbound} on the nonlinearity is independent of $\epsilon$,
there exists $M(\ell,r,\beta)>0$ such that 
$\|z_{\epsilon,\infty}\|_{H_0^{9/8}}\le M(\ell,r,\beta)$
for all $\epsilon\ge 0$.
\end{proof}

\begin{corollary}\label{cor:eps}
$z_{\epsilon,\infty}\to z_\infty$ as $\epsilon\to 0$ in $H^1_0(\Omega)$. 
Hence the expressions for $\ell_t$ and $r_t$ in
equation~\eqref{eqn:reduced} for $\epsilon>0$ converge as $\epsilon\to 0$ 
to the corresponding expression for $\epsilon=0$.
\end{corollary}

\begin{proof}
For notational convenience let us write $z_{\epsilon}=z_{\epsilon,\infty}$.  
First we establish convergence in $L^{\infty}(\Omega)$. From the weak 
comparison principle, we get that if $0\leq\tilde{\epsilon}\leq\epsilon$ 
then $z_{\epsilon}\leq z_{\tilde{\epsilon}}$ and 
$z_{\tilde{\epsilon}}+\tilde{\epsilon}\leq z_{\epsilon}+\epsilon$.  
To see this note that for $\xi_{\epsilon}=z_{\epsilon}+\epsilon$ and 
$\xi_{\tilde{\epsilon}}=z_{\tilde{\epsilon}}+\tilde{\epsilon}$ we have
\begin{gather*} 
\frac{4}{9} \partial^2_{xx}(\xi_{\tilde{\epsilon}}-\xi_{\epsilon})
+(r-l)^2\frac{2}{3}F((r-l)x+l)\Big(\frac{1}{\xi_{\tilde{\epsilon}}^{5/3}}
-\frac{1}{\xi_{\epsilon}^{5/3}}\Big)=0\\
-\frac{\beta}{3}\Big(\frac{1}{\xi_{\tilde{\epsilon}}^{1/3}}
-\frac{1}{\xi_{\epsilon}^{1/3}}\Big) \quad \text{in $\Omega$},\\
   z_{\tilde{\epsilon}}-z_{\epsilon}\leq 0\quad \text{on $\partial\Omega$}.
  \end{gather*}
From this it follows that $(z_{\epsilon})_{\epsilon>0}$ is a Cauchy 
sequence in $L^{\infty}(\Omega)$ and there exists $z\in L^{\infty}(\Omega)$ 
satisfying $z_{\epsilon}\to z$ as $\epsilon\to 0$ in $L^{\infty}(\Omega)$ 
and $\underline{z}=\lim_{\epsilon\to 0^+}\underline{z}_{\epsilon}\leq z\leq \bar{z}$.
From Theorem \ref{stable} we have that $z_{\epsilon,\infty}\to z$ in 
$H^1_0(\Omega)$. For $\varphi\in H^1_0(\Omega)$
\begin{align*}
&\int_{\Omega}\left(z_\epsilon\right)_{xx}\varphi dx
+\frac{(r-\ell)^2}{3}\int_{\Omega}\Big(\frac{2F((r-\ell)x+\ell)}
{ z_{\epsilon}^{5/3}}-\frac{\beta}{ z_{\epsilon}^{1/3}}\Big)\\
&=-\int_{\Omega}\left(z_\epsilon\right)_{x}\varphi_{x} dx
 +\frac{(r-\ell)^2}{3}\int_{\Omega}
 \Big(\frac{2F((r-\ell)x+\ell)}{ z_{\epsilon}^{5/3}}
 -\frac{\beta}{ z_{\epsilon}^{1/3}}\Big)=0.
\end{align*}
Letting $\epsilon\to 0$ yields
\[
-\int_{\Omega}z_{x}\varphi_{x} dx+\frac{(r-\ell)^2}{3}
\int_{\Omega}\Big(\frac{2F((r-\ell)x+\ell)}{ z^{5/3}}
 -\frac{\beta}{z^{1/3}}\Big)=0.
\]
Therefore $z$ is a weak solution to the stationary equation for \eqref{Pt},
and since $z_\infty$ is the unique such solution, we have 
$z_{\epsilon,\infty}\to z_\infty$ as $\epsilon\to 0$.
Since the right hand side of \eqref{eqn:reduced} converges pointwise a.e.
in $\Omega$, letting $\epsilon\to 0$ and applying the Dominated 
Convergence Theorem yields the convergence of $\ell_t$ and $r_t$.
\end{proof}

Next we establish that the maximal time interval of existence  for  
\eqref{Ste} relies only on the dynamics of $\ell_{\epsilon}(t)$ and 
$r_{\epsilon}(t)$.

\begin{theorem}\label{thm:cone}
The solution $(z_{\epsilon}(t),l_{\epsilon}(t),r_{\epsilon}(t))$ to \eqref{Ste} 
stays in $\mathsf{C}(\ell,r,\beta,\epsilon)\times I^2$ for all $t\ge 0$ in 
its maximal interval of existence.
Moreover, either the solution can be extended globally to $[0,\infty)$, 
or at the maximal time of existence
$T$, either $\ell_\epsilon(T)=u_{\rm min}$ and $\ell$ is decreasing at time 
$T$ or $r_\epsilon(T)=u_{\rm max}$ and $r$ is increasing at time $T$, i.e.\ either the
maximal interval of existence is infinite, or either $\ell_\epsilon$ 
or $r_\epsilon$ leaves $I$ in finite time.
\end{theorem}

\begin{proof}
Suppose that $\ell_{\epsilon}(t)$, $r_{\epsilon}(t)$
remain in $I$ over the time interval $[0,T]$.
From Theorem \ref{stable} the set  
$\{(z_{\epsilon}(t),\ell_{\epsilon}(t),r_{\epsilon}(t))\}$ 
is compact in $C\left([T/2,T], H^{1}_{0}(\Omega)\right)\times I^2$.
We can then choose a sequence 
$\{(z_{\epsilon}(t_n),\ell_{\epsilon}(t_n),r_{\epsilon}(t_n))\}_{n\geq1}$ 
such that $(z_{\epsilon}(t_n),\ell_{\epsilon}(t_n),r_{\epsilon}(t_n))
\to(z^{*}_{\epsilon},\ell_{\epsilon}^{*},r_{\epsilon}^{*})$ for $t_n\to T$.
Treating $(z^{*}_{\epsilon},\ell_{\epsilon}^{*},r_{\epsilon}^{*})$ as new 
initial conditions, if $\ell$ is nondecreasing and $r$ is nonincreasing 
at time $T$, the solution can be extended beyond time $T$.
Therefore, if the maximal interval of existence is $[0,T]$ with $T<\infty$,
we must have either $\ell(T)=u_{\rm min}$ with $\ell$ decreasing at $T$
or $r(T)=u_{\rm max}$ with $r$ increasing at $T$.
 \end{proof}

 The following proposition shows that in the regularized system with 
$\epsilon>0$, the  integral expressions for $\ell_t, r_t$ in 
equation~\eqref{eqn:reduced} have a simpler pointwise formulation.

\begin{proposition}\label{prop:ptwise}
Let $z_{\epsilon,\infty}$ be the family of solutions of $E_\epsilon[z]=0$ 
for $\epsilon>0$. Recalling
that $u=(r-\ell)x+\ell$, then
\begin{gather}\label{l_t}
\partial_{\ell}J^{\epsilon}[z,\ell,r]
=\frac{2}{9}\left(\partial_{u}z_{\epsilon,\infty}(\ell)\right)^2
-\frac{F(\ell)}{\epsilon^{2/3}}-\frac{\beta}{2}\epsilon^{2/3}, \\
\label{r_t}
\partial_{r}J^{\epsilon}[z,\ell,r]=-\frac{2}{9}
\left(\partial_{u}z_{\epsilon,\infty}(r)\right)^2+
\frac{F(r)}{\epsilon^{2/3}}+\frac{\beta}{2}\epsilon^{2/3}.
\end{gather}
\end{proposition}

\begin{proof}
We start by reducing the  expression
\begin{align*}
&\int_{\Omega}\frac{2}{9(r-\ell)^2}z_x^2
 +\frac{(r-\ell)(1-x)F'-F}{(z+\epsilon)^{2/3}}
 -\frac{\beta}{2}(z+\epsilon)^{2/3} dx\\
&=\int^{r}_{\ell}\frac{2}{9(r-\ell)}z_u^2+\frac{F_{\ell}}{(z+\epsilon)^{2/3}}
 -\frac{F}{(r-\ell)(z+\epsilon)^{2/3}}
 -\frac{\beta (z+\epsilon)^{2/3}}{2(r-\ell)} du.
\end{align*}
The term $\int^{r}_{\ell}\frac{F_{\ell}}{(z+\epsilon)^{2/3}}du$ can be 
further simplified
\begin{align*}
&\int^{r}_{\ell}\frac{F_{\ell}}{(z+\epsilon)^{2/3}}du \\
&=\int^{r}_{\ell}\frac{1}{(z+\epsilon)^{2/3}}\frac{du}{d\ell}\frac{dF}{du}du
 =\frac{1}{(r-\ell)}\int^{r}_{\ell}\frac{r-u}{(z+\epsilon)^{2/3}}\frac{dF}{du}du\\
&=\frac{(r-u)}{(r-\ell)}\frac{F}{(z+\epsilon)^{2/3}}\Big|^{r}_{\ell}
 -\frac{1}{(r-\ell)}\int^{r}_{\ell}\frac{d}{du}
 \big[\frac{r-u}{(z+\epsilon)^{2/3}}\big]Fdu\\
&=-\frac{F(\ell)}{\epsilon^{2/3}}+\frac{1}{(r-\ell)}\int^{r}_{\ell}
\frac{F}{(z+\epsilon)^{2/3}}du
+\frac{2}{3(r-\ell)}\int^{r}_{\ell}\frac{(r-u)F}{(z+\epsilon)^{5/3}}z_{u}du.
\end{align*}
Next note that
\begin{align*}
&\frac{2}{3(r-\ell)}\int^{r}_{\ell}F\frac{(r-u)}{(z+\epsilon)^{5/3}}z_{u}du\\
&=-\frac{4}{9(r-\ell)}\int^{r}_{\ell}(r-u)z_{uu}z_{u}du 
 +\frac{\beta}{3(r-\ell)}\int^{r}_{\ell}(r-u)(z+\epsilon)^{-1/3}z_{u}du\\
&=-\frac{2}{9(r-\ell)}\int^{r}_{\ell}(r-u)\frac{d}{du}(z_u^2)du 
 +\frac{\beta(r-u)}{2(r-\ell)}(z+\epsilon)^{2/3}\Big|^{r}_{\ell}
 +\frac{\beta}{2(r-\ell)}\int^{r}_{\ell}(z+\epsilon)^{2/3} du\\
&=\frac{2}{9}z_{u}^2(\ell)-\frac{2}{9(r-\ell)}\int^{r}_{\ell}z_u^2du
 -\frac{\beta}{2}\epsilon^{2/3}+\frac{\beta}{2(r-\ell)}
 \int^{r}_{\ell}(z+\epsilon)^{2/3}du.
\end{align*}
Therefore
\begin{align*}
&\int_{\Omega}\frac{2}{9(r-\ell)^2}z_x^2+\frac{(r-\ell)(1-x)F'-F}{(z+\epsilon)^{2/3}}
 -\frac{\beta}{2}(z+\epsilon)^{2/3} dx\\
&=\frac{2}{9}z_{u}^2(\ell)-\frac{F(\ell)}{(z+\epsilon)^{2/3}}
 -\frac{\beta}{2}(z+\epsilon)^{2/3}\\
 &=\frac{2}{9}z_{u}^2(\ell)-\frac{F(\ell)}{\epsilon^{2/3}}
 -\frac{\beta}{2}\epsilon^{2/3},
\end{align*}
the last equality follows from the boundary conditions $z(\ell)=z(r)=0$.
The proposition now follows from Equation~\eqref{eqn:fullgradient}.
\end{proof}

Note that Corollary~\ref{cor:eps} and Proposition~\ref{prop:ptwise}
can be used together to establish the signs of $\ell_t$ and $r_t$
in equation~\eqref{eqn:reduced} at $\epsilon=0$.
In Corollary~\ref{cor:improved} we established the existence of simple closed characteristics in the more
general setting of Proposition~\ref{prop:lem9}. We now use the results of this section to describe a
framework to consider non-simple closed characteristics.


\section{Final remarks: a framework for non-simple closed characteristics}
\label{sec:example}

In the previous sections we built a framework in which to study a gradient-like 
dynamics using the second-order Lagrangian action functional
\begin{equation} \label{nsccL}
J[u]=\int_0^\tau  \frac{1}{2}(u'')^2+\frac{\beta}{2} (u')^2+F(u)\,dt
\end{equation}
where we first considered an individual monotone lap $u$ with $u(0)=\ell, u(\tau)=r$
and $u'>0$ in $(0,\tau)$.
Now we consider a periodic function $u$ as a concatenation of finitely many laps,
which yields a closed curve in the $(u,u')$-plane.
Recall that for each increasing lap with $u(0)=\ell,\,u(\tau)=r$, and $\ell<r$,
we define $z=|u'|^{3/2}$, and $z$ is a function of $u$ by monotonicity.
Changing variables, $z$ is taken to be a function
of the variable $x$ defined by $u=(r-\ell)x+\ell$. The action functional
$J[z;\ell,r]$ in these variables is given in equation~\eqref{actionJ}.
In the case of a decreasing lap with $u(0)=r$, $u(\tau)=\ell$, and
$\ell<r$, we still define $z=|u'|^{3/2}$ so that $z$ is a function of $u$
from $\ell$ to $r$, and hence  the variable $x$ is again defined by
$u=(r-\ell)x+\ell$, and $J[z;r,\ell]$  in these variables is given again
by equation~\eqref{actionJ}.
Indeed, since the Lagrangian $L$ has the symmetry $L(u,-v,w)=L(u,v,w)$,
we have that if $u(t)$ is an increasing lap from $\ell$ to $r$, then
$u(\tau-t)$ is a decreasing lap from $r$ to $\ell$ so that $z$ is the same
function  in both cases, and hence $J[z;\ell,r]=J[z;r,\ell]$.

The gradient dynamics of $J$ for functions with $n$ laps can then be expressed 
as a coupled system of $n$ parabolic equations and $n$ ODE's by extending 
the system \eqref{Ste} defined in Section~\ref{sec:EE}.
The resulting dynamics can be viewed as a type of curve evolution in the 
$(u,u')$-plane. However, as forecast in the previous section, we reduce this 
dynamics by fixing the laps to be in the family of unique  stationary
laps determined by fixed endpoints in Theorem~\ref{stable}. 
This results an $n$-dimensional system of ODE's for the endpoints only, 
and stationary solutions of the reduced system must be stationary for the 
full system.

We now derive  the system we consider in some detail.
As described in Section~\ref{sec:prelim}, the method of broken geodesics has been
applied to second-order Lagrangian systems in \cite{RV}  in the case where 
unique laps exist as minimizers as follows.
Let $u_1,u_2,\ldots,u_n$ and $u_{n+1}=u_1$
be the local extreme values of $u$ and define
$$S(u_k,u_{k+1})=\min J[z;u_k,u_{k+1}]=J[(z_\infty)_k,u_k,u_{k+1}].$$
Then define $W(u_1,\ldots,u_n)=\sum_{i=1}^n S(u_k,u_{k+1})$ which is the sum of the
actions of the  minimizing laps, i.e.~the action of the broken geodesic.
The $k$-th component of the gradient of $W$ is simply 
 $\partial_2S(u_{k-1},u_{k})+\partial_1S(u_{k},u_{k+1})$. Properties of these 
derivatives can be exploited to prove the existence of critical points for $W$, 
which yield closed characteristics, see \cite{RV}.

Here we let $z_k$ be the variable of the $k$-th lap and consider  the action of the
broken geodesic which is given by
$$
\widehat{W}[z_1,\ldots,z_n; u_1,\ldots,u_n]=\sum_{k=1}^n J[z_k,u_k,u_{k+1}].
$$
Assume that $u_1<u_2$ so that for $k$ odd $u_k$ is a minimum and for $k$ even
$u_k$ is a maximum of the broken geodesic.
Then from equation \eqref{eqn:fullgradient}
we have
\begin{equation}\label{eq:endpt}
\begin{aligned}
&(-1)^{k+1}\partial_{u_k}\widehat{W} \\
&=\frac{2}{9(u_{k-1}-u_k)^2}\int_{\Omega}(\partial_{x}z_{k-1})^2\,dx
+\frac{2}{9(u_{k+1}-u_k)^2}\int_{\Omega}(\partial_{x}z_{k})^2\,dx\\
&+ \int_{\Omega}\frac{(u_{k-1}-u_{k})\sigma_k(x)F'((u_{k-1}-u_{k})x
 +u_{k})-F((u_{k-1}-u_{k})x+u_{k})}{z_{k-1}^{2/3}}\,dx\\
&+\int_{\Omega}\frac{(u_{k+1}-u_k)\sigma_k(x)F'((u_{k+1}-u_{k})x+u_{k})
 -F((u_{k+1}-u_{k})x+u_{k})}{z_{k}^{2/3}}\,dx\\
&-\frac{\beta}{2}\int_\Omega z_{k-1}^{2/3}\,dx
-\frac{\beta}{2}\int_\Omega z_{k}^{2/3}\, dx
\end{aligned}
\end{equation}
where $\sigma_k(x)=1-x$ for $k$ odd and $\sigma_k(x)=-x$ for $k$ even.

Substituting $(z_\infty)_k$ into the gradient-like
system derived from equation \eqref{eq:endpt} yields a dynamics for the 
evolution of the endpoints of the closed curve, as in Section~\ref{sec:EE}. 
 One can simplify the terms involving $\int_{\Omega}(\partial_{x}z_{k})^2\,dx$
in \eqref{eq:endpt}.
Using the fact that $(z_\infty)_k$ is a solution to the stationary 
PDE \eqref{ellipticEq} we have 
\[ %\label{eq:eq}
\frac{2}{9(u_{k+1}-u_k)^2}\int_\Omega(\partial_x(z_\infty)_k)^2\,dx
=\frac{1}{3}\int_\Omega\frac{F((u_{k+1}-u_k)x+u_k)}{(z_\infty)_k^{2/3}}\,dx
-\frac{\beta}{6}\int_\Omega (z_\infty)_k^{2/3}\,dx.
\]

Moreover, to obtain a pointwise estimate of the right-hand side of \eqref{eq:endpt} 
one can consider the corresponding perturbed system and apply Proposition 
\ref{prop:ptwise}. This would lead to pointwise estimates for the unperturbed 
system after sending $\epsilon\to 0$ and applying Corollary \ref{cor:eps} 
along with standard convergence theorems.  Furthermore using the explicit 
upper and lower solutions from Section \ref{subsec:lus} the integral terms 
in \eqref{eq:endpt} could be numerically investigated.  This type of analysis 
could be helpful in constructing an index pair or isolating neighborhood 
for the classes of non-simple closed characteristics with certain profiles.

\begin{thebibliography}{10}

\bibitem{RA} Ronald Adams;
 \emph{Curve {S}hortening in {S}econd-{O}rder {L}agrangian
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\end{thebibliography}

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