\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 61, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2011/61\hfil Classification of heteroclinic orbits]
{Classification of heteroclinic orbits of semilinear
parabolic equations with a polynomial nonlinearity}
\author[M. Robinson\hfil EJDE-2011/61\hfilneg]
{Michael Robinson} % in alphabetical order
\address{Michael Robinson \newline
Mathematics Department\\
University of Pennsylvania\\
209 S. 33rd Street\\
Philadelphia, PA 19104, USA}
\email{robim@math.upenn.edu}
\thanks{Submitted August 26, 2010. Published May 10, 2011.}
\subjclass[2000]{35B40, 35K55}
\keywords{Heteroclinic connection; semilinear parabolic equation;
equilibrium}
\begin{abstract}
For a given semilinear parabolic equation with polynomial
nonlinearity, many solutions blow up in finite time. For a certain
class of these equations, we show that some of the solutions which do
not blow up actually tend to equilibria. The characterizing property
of such solutions is a finite energy constraint, which comes about
from the fact that this class of equations can be written as the flow
of the $L^2$ gradient of a certain functional.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{calculation}[theorem]{Calculation}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}
Dynamics of semilinear parabolic equations play an important role in
certain applications, and have a long history of study. Perhaps the
earliest occurrence is in a pair of articles
\cite{Fisher_1937, KPP_1937}, where the long-time behavior of
solutions is addressed in the context of population biology. The
behavior of global solutions that approach equilibria for positive and
\emph{negative} time is also of special interest. These
\emph{heteroclines} capture the admissible transitions between
equilibria, which is important for assembling a dynamical picture
of spaces of solutions.
In this article, the global behavior of smooth solutions to the
semilinear parabolic equation
\begin{equation}\label{pde}
\frac{\partial u(t,x)}{\partial t}=\Delta u(t,x) - u^N
+ \sum_{i=0}^{N-1} a_i(x) u^i(t,x)=\Delta u + P(u),
\end{equation}
%
for $(t,x) \in \mathbb{R}\times\mathbb{R}^n = \mathbb{R}^{n+1}$ is
considered, where $N\ge 2$ and $a_i \in L^\infty(\mathbb{R}^n)$ are
smooth with all derivatives of all orders bounded. It suffices to
note that the existence of multiple equilibria (see
\cite{RobinsonNonauto}) quickly foils any hope for a common limit for
all global solutions, and paves the way for more complicated dynamics.
\subsection{Article highlights}
Our main result (Theorem \ref{limits_to_equilibria}) is that
heteroclinic orbits of \eqref{pde} connecting two finite action
equilibrium solutions (Definition \ref{action_df}) are characterized
by finite energy (Definition \ref{energy_df}). (Time limits of
solutions will be understood in the sense of uniform convergence on
compact subsets.) That this characterization is necessary at all comes
from the fact that the spatial domain of \eqref{pde} is unbounded.
For bounded spatial domains, all bounded global solutions converge to
equilibria. \cite{Jost_2007} The strength of our result comes from the
fact that the finite energy constraint makes solutions behave rather
well. Therefore, our result is much sharper than what has typically
been obtained in the past, and it applies to more complicated
nonlinear terms.
Heteroclinic orbits of \eqref{pde} are rare: that there are any such
solutions at all is shown in \cite{RobinsonGlobal}. Under certain
conditions on the $a_i$, the space of heteroclinic orbits is a
finite-dimensional cell complex \cite{RobinsonThesis}, contained in an
time-weighted (but not spatially-weighted) Sobolev space. As an
example of the rarity of heteroclines, we note in passing that
travelling wave solutions do not have finite energy. Even though a
travelling wave will often converge locally to equilibria, at least
one of those equilibria will not have finite action. On the other
hand, we can exclude travelling waves from the solution set of
\eqref{pde} if we require that all the coefficients $a_i$ decay fast
enough and only consider one spatial dimension. Then our result
establishes an equivalence between all heteroclinic orbits and the
finite energy solutions, as all equilibria have finite action. (See
\cite{RobinsonNonauto} for a demonstration that decay of the $a_i$
implies finite action.)
A crucial requirement in this article is that the equilibria be
\emph{isolated}. While it is unlikely that the equilibria of \eqref{pde}
are always isolated, they are in many interesting cases. We therefore
examine some sufficient conditions for isolatedness of equilibria
(Lemma \ref{injective_schroedinger}).
\subsection{Historical comments}
The study of \eqref{pde} on unbounded domains is not new. Blow-up
behavior for equations like \eqref{pde} was examined in a classic
paper by Fujita. \cite{Fujita} This line of classical reasoning was
studied by many authors, and is summarized in \cite{Zheng_1995}. For
somewhat more restricted nonlinearities, Du and Ma were able to use
squeezing methods to obtain similar results to what we obtain here.
In particular, they also show that certain kinds of solutions approach
equilibria. \cite{DuMa2001} A major difference between the results
obtained by Du and Ma and the work presented here is that in our case
there is a lack of uniqueness, both in the global solutions themselves
and also in their limits. Generally speaking, there will be many
equilibrium solutions to which global solutions may tend, each with
different dynamical properties.
In a somewhat different setting, Floer used a finite energy constraint
for solutions and a regularity constraint on equilibria to
characterize heteroclinic orbits of an elliptic
problem. \cite{Floer_gradient} The techniques of Floer were
subsequently used by Salamon to provide a new characterization of
solutions to gradient flows on finite-dimensional
manifolds. \cite{Salamon_1990} In this article, we recast some of
Salamon's work into a parabolic setting, and of course work within an
infinite-dimensional space.
\subsection{Outline of the article}
In Section \ref{defs_sec}, we present definitions of \emph{energy} and
\emph{action} for solutions to \eqref{pde} and timeslices of solutions,
respectively. Our characterization theorem is proven in Section
\ref{conv_sec}, and discussed in Section \ref{disc_sec}.
\section{Finite energy constraints}
\label{defs_sec}
It is well-known that solutions to \eqref{pde} exist along strips of
the form $(t,x)\in I\times \mathbb{R}^n$ for sufficiently small,
positive $t$-intervals $I$. One might hope to extend such solutions
to all of $\mathbb{R}^{n+1}$, but for certain choices of initial
conditions such global solutions may fail to exist. \cite{Fujita} We
will specifically avoid blow-up by considering only global solutions
to \eqref{pde}. By global solutions, we mean those which are defined
for all $\mathbb{R}^{n+1}$, have one continous partial derivative in
time, and two continous partial derivatives in space. It should be
noted that global solutions to \eqref{pde} are quite rare: the
backwards-time Cauchy problem contains a heat operator, and so most
solutions will not extend to all of $\mathbb{R}^{n+1}$. Existence of
global solutions (as in \cite{RobinsonGlobal}) is therefore a
feature of the nonlinear term in \eqref{pde}.
\begin{definition} \label{action_df} \rm
Our analysis of \eqref{pde} will make considerable use of the fact
that it is a gradient differential equation. Observe that the right
side of \eqref{pde} is the $L^2(\mathbb{R}^n)$ gradient of the
following \emph{action functional}, defined for all $f \in
C^1(\mathbb{R}^n)$:
%
\begin{equation}
\label{action_eqn}
A(f)=\int_{\mathbb{R}^n} - \frac{1}{2} \|\nabla f(x)\|^2
- \frac{f^{N+1}(x)}{N+1}+ \sum_{i=0}^{N-1}
\frac{a_i(x)}{i+1} f^{i+1}(x) dx.
\end{equation}
\end{definition}
%
It is then evident that along a solution
$u(t) \in L^2(\mathbb{R}^n)$ to \eqref{pde},
\begin{align*}
\frac{dA(u(t))}{dt}
&= dA|_{u(t)}\big(\frac{\partial u}{\partial t}\big)\\
&= \langle \nabla A(u(t)), \frac{\partial
u}{\partial t} \rangle\\
&= \langle \Delta u + P(u), \frac{\partial
u}{\partial t} \rangle\\
&= \|\frac{\partial
u}{\partial t} \|_2^2 \ge 0,\\
\end{align*}
so $A(u(t))$ is a monotone function. As an immediate consequence,
nonconstant $t$-periodic solutions to \eqref{pde} do not exist.
\begin{definition}\label{energy_df} \rm
The \emph{energy functional} is the following quantity defined on the
space $S$ of functions $\mathbb{R}^{n+1}\to \mathbb{R}$ with one
continuous partial derivative in the first variable ($t$), and two
continuous partial derviatives in the rest ($x$):
\begin{equation}
\label{energy_eqn}
E(u)=\frac{1}{2}\int_{-\infty}^\infty \int
| \frac{\partial u}{\partial t} |^2 + | \Delta u + P(u) |^2 dx\, dt.
\end{equation}
It is evident that $S$ is a Banach space under the appropriate norm,
which is
\[
\|u\|_S=\|u\|_\infty+\|\frac{\partial u}{\partial t}
\|_\infty+\sum_{i=1}^n \|\frac{\partial u}{\partial x_i}
\|_\infty+\sum_{i,j=1}^n\| \frac{\partial^2 u}{\partial x_i
\partial x_j}\|_\infty.
\]
\end{definition}
\begin{calculation} \label{action_energy_calc} \rm
Suppose $u \in S$ is in the domain of definition for the energy
functional, then
%
\begin{align*}
E(u)&= \frac{1}{2}\int_{-\infty}^\infty \int | \frac{\partial u}{\partial t}
|^2 + | \Delta u + P(u) |^2 dx\, dt\\
&= \frac{1}{2}\int_{-\infty}^\infty \int
\Big( \frac{\partial u}{\partial t}
- \Delta u - P(u) \Big)^2 + 2\frac{\partial u}{\partial t}
( \Delta u + P(u)) dx\, dt\\
&= \frac{1}{2}\int_{-\infty}^\infty \int
\Big( \frac{\partial u}{\partial t}
- \Delta u - P(u) \Big)^2 dx\, dt
+ \int_{-\infty}^\infty \langle \frac{\partial
u}{\partial t}, \Delta u + P(u) \rangle dt\\
&= \frac{1}{2}\int_{-\infty}^\infty \int
\Big( \frac{\partial u}{\partial t}
- \Delta u - P(u) \Big)^2 dx\, dt
+ \int_{-\infty}^\infty \langle \frac{\partial
u}{\partial t}, \nabla A(u(t))\rangle dt\\
&= \frac{1}{2}\int_{-\infty}^\infty \int
\Big( \frac{\partial u}{\partial t}
- \Delta u - P(u) \Big)^2 dx\, dt + \int_{-\infty}^\infty
\frac{d}{dt} A(u(t)) dt\\
&= \frac{1}{2}\int_{-\infty}^\infty \int
\Big( \frac{\partial u}{\partial t}
- \Delta u - P(u) \Big)^2 dx\, dt + A(u(T)) \Big|_{T=-\infty}^\infty.
\end{align*}
%
This calculation shows that finite energy solutions to \eqref{pde}
minimize the energy functional over functions in $S$ with $t$-boundary
conditions being equilibria of \eqref{pde}, and $x$-boundary
conditions enforced by finiteness of the integrals. If a solution to
\eqref{pde} is a heteroclinic connection between two equilibria, then
the energy functional measures the difference between the values of
the action functional evaluated at the two equilibria. \emph{The main
result of this article is the converse, that finite energy
characterizes the solutions which connect equilibria.}
\end{calculation}
Finite energy solutions to \eqref{pde} are even more rare than global
solutions. However, the set of finite energy solutions is not entirely
vacuous, as equilibrium solutions automatically have finite energy.
Not every equation of the form \eqref{pde} will have equilibria, but
some do. Consider
\[
\frac{\partial u}{\partial t} = \Delta u - u^2,
\]
which evidently has the zero function as an equilibrium. Indeed, the
zero function is the \emph{only} finite energy solution \cite{Fujita}.
It is well-known that equations like \eqref{pde} exhibiting
translational symmetry in space may support travelling wave solutions
of the form $u(t,x)=U(x-ct)$ for some $c \in
\mathbb{R}$. \cite{FiedlerScheel} As a result, it is immediate that
travelling waves will have infinite energy. On the other hand, they
also evidently connect equilibria (as measured using the topology of
uniform convergence on compact subsets, as opposed to the topology of
$S$ defined earlier). Calculation \ref{action_energy_calc} shows that
a necessary condition for travelling waves is that there exists at
least one equilibrium whose action is infinite. In this article, we
will consider only equilibria with finite action, and solutions with
finite energy.
\section{Convergence to equilibria}
\label{conv_sec}
In this section, we show that finite energy solutions tend to
equilibria as $|t| \to \infty$, culminating in the proof of the
following theorem.
\begin{theorem} \label{limits_to_equilibria}
Suppose that either $n=1$ or $N$ is odd and that equilibria are
isolated. A smooth global solution $u$ to \eqref{pde} has finite
energy if and only if each of the following hold:
\begin{itemize}
\item each of $U_{\pm}(x) =\lim_{t \to
\pm \infty} u(t,x)$ exists and converges with its first derivatives
uniformly on compact subsets of $\mathbb{R}^n$,
\item $U_{\pm}$ are bounded, continuous equilibrium solutions to \eqref{pde},
\item and either $|A(U_+)-A(U_-)|<\infty$ or $U_+=U_-$.
\end{itemize}
\end{theorem}
We follow Floer in \cite{Floer_gradient} which leads us through an
essentially standard parabolic bootstrapping argument. We begin,
however, with a result that is consequence of the fact that
$W^{k,\infty}(\mathbb{R}^{n+1})$ is a Banach algebra. This permits a
straightforward treatment of the polynomial nonlinearity in
\eqref{pde}.
\subsection*{Convention}
We employ the standard multi-index notation $D^k$ to refer to the set
of order $k$ derivative operators, to be taken over the principal
directions in $\mathbb{R}^{n+1}$. For instance, $D^1$ refers to the
set $\{\frac{\partial}{\partial t}, \frac{\partial}{\partial x_1},
\frac{\partial}{\partial x_2}, \dots \frac{\partial}{\partial x_n}\}$.
\begin{lemma}\label{polybound_lem}
Let $U \subseteq \mathbb{R}^n$ and $u \in W^{k,p}(U)$ satisfy $\|D^j
u\|_\infty \le C < \infty$ for $0 \le j \le k$ (in particular, $u$ is
bounded). If $P(u) = \sum_{i=1}^N a_i u^i$ with $a_i \in L^\infty(U)$
then there exists a $C'$ such that $\|P(u)\|_{k,p} \le C'
\|u\|_{k,p}$. (Recall that we also assume that $\|D^j a_i\|_\infty$
exist and are all finite.)
\end{lemma}
We note that this is a local result, and therefore does not require
decay conditions on the $a_i$.
\begin{proof}
First, using the definition of the Sobolev norm,
\[
\|P(u)\|_{k,p}=\sum_{j=0}^k \|D^j P(u) \|_p \le \sum_{j=0}^k
\sum_{i=1}^N \|D^j a_i u^i \|_p.
\]
Now $|D^j a_i u^i| \le P_{i,j}(u,Du,\dots ,D^j u)$, which is
a polynomial in $j$ variables with constant coefficients,
and no constant term.
(The constant coefficients is a consequence of the bounded derivatives
of the $a_i$.) Additionally,
\begin{align*}
\|(D^m u)^q D^j u\|_p &= \Big( \int | (D^m u)^q D^j u |^p \Big)^{1/p}\\
&\le \|D^m u\|_\infty^q \Big( \int | D^j u |^p \Big)^{1/p}
\le C^q \|D^j u\|_p,
\end{align*}
so by collecting terms,
\[
\|P(u)\|_{k,p} \le \sum_{j=0}^k
\sum_{i=1}^N \|D^j a_i u^i \|_p \le \sum_{j=0}^k A_j \|D^j u\|_p
\le C' \|u\|_{k,p}.
\]
\end{proof}
The following result is a parabolic bootstrapping argument that does
most of the work. In it, we follow Floer in \cite{Floer_gradient},
replacing ``elliptic'' with ``parabolic'' as necessary.
\begin{lemma} \label{parabolic_bootstrap_lem}
Suppose that all of the equilibria of \eqref{pde} are isolated. If
$u$ is a finite energy solution to \eqref{pde} with $\|D^j
u\|_{L^\infty((-\infty,\infty)\times V)} \le C < \infty$ for $0 \le j
\le k$ with $k\ge 1$ on each compact $V \subset \mathbb{R}^n$, then
each of $\lim_{t \to \pm \infty} D^j u(t,x)$ exists, and each converges
uniformly on compact subsets of $\mathbb{R}^n$. Further, the limits
are equilibrium solutions to \eqref{pde}. (Again, we assume
$\|D^ja_i\|_\infty<\infty$ as before.)
\end{lemma}
\begin{proof}
Define $u_m(t,x)=u(t_m,x)$ where $t_m \to \infty$. Suppose $U \subset
\mathbb{R}^{n+1}$ is a bounded open set and $K \subset U$ is compact.
Let $\beta$ be a smooth bump function whose support is $\bar{U}$,
takes the value 1 on $K$, and is nonzero within $U$. We take $p>1$
such that $kp>n+1$. Then we can consider $u_m \in W^{k,p}(U)$ (recall
that $u$ and its first $k$ derivatives of $u$ are bounded on the
closure of $U$), and we have
\[
\|u_m\|_{W^{k+1,p}(K)} \le \|\beta u_m \|_{W^{k+1,p}(U)}.
\]
Then using the standard regularity for the parabolic operators,
\[
\|\beta u_m \|_{W^{k+1,p}(U)} \le C_1 \big\| \Big(
\frac{\partial}{\partial t} - \Delta +
\frac{2}{\beta}\nabla\beta \cdot \nabla \Big) (\beta u_m)
\big\|_{W^{k,p}(U)}.
\]
The usual product rule yields the following:
\[
\Big(\frac{\partial}{\partial t} - \Delta +
\frac{2}{\beta}\nabla\beta \cdot \nabla \Big)(\beta u_m) =
u_m \Big(\frac{\partial}{\partial t} - \Delta +
\frac{2}{\beta}\nabla\beta \cdot \nabla \Big) \beta +
\beta\Big(\frac{\partial}{\partial t} - \Delta \Big)u_m
\]
%
which implies that
%
\begin{align*}
&\big\|\Big(\frac{\partial}{\partial t} - \Delta +
\frac{2}{\beta}\nabla\beta \cdot \nabla\Big)(\beta
u_m)\|_{W^{k,p}(U)}\\
&\le \big\|u_m \Big(\frac{\partial}{\partial t} - \Delta +
\frac{2}{\beta}\nabla\beta \cdot \nabla\Big) \beta
\|_{W^{k,p}(U)}
+ \big\|\beta\Big(\frac{\partial}{\partial t} - \Delta
\Big)u_m\big\|_{W^{k,p}(U)}.
\end{align*}
Let $P'(u)=-u^N+\sum_{i=1}^{N-1} a_i u^i$, noting carefully that we have left
out the $a_0$ term. Hence, as suggested in \cite{Salamon_1990}
we obtain
\begin{align*}
&\|u_m\|_{W^{k+1,p}(K)}\\
&\leq C_1 \|\beta
\Big(\frac{\partial}{\partial t} - \Delta \Big) u_m
\|_{W^{k,p}(U)} + C_2 \|u_m\|_{W^{k,p}(U)}\\
&\leq C_1 \|\beta
\Big(\frac{\partial}{\partial t} - \Delta \Big) u_m + \beta P'(u_m)
- \beta P'(u_m) \|_{W^{k,p}(U)} + C_2 \|u_m\|_{W^{k,p}(U)}\\
&\leq C_1 \|\beta a_0\|_{W^{k,p}(U)} + C_1 \| \beta P'(u_m)
\|_{W^{k,p}(U)} + C_2 \|u_m\|_{W^{k,p}(U)}\\
&\leq C_1 \|\beta a_0\|_{W^{k,p}(U)} + C_3 \|u_m\|_{W^{k,p}(U)},
\end{align*}
where the last inequality is a consequence of Lemma
\ref{polybound_lem}. By the hypotheses on $u$ and $a_0$, this implies
that there is a finite bound on $\|u_m\|_{W^{k+1,p}(K)}$, which is
independent of $m$. Now by our choice of $p$, the general
Sobolev inequality implies that $\|u_m\|_{C^{k+1-(n+1)/p}(K)}$ is
uniformly bounded. By choosing $p$ large enough, there is a
subsequence $\{v_{m'}\} \subset \{u_m\}$ such that $v_{m'}$ (and its
first $k$ derivatives) converge uniformly on $K$, say to $v$. For any
$T>0$, we observe
\begin{align*}
\int_{-T}^T \int | \frac{\partial v}{\partial t} |^2 dx\, dt
&= \lim_{{m'} \to \infty}\int_{-T}^T \int | \frac{\partial v_{m'}}{\partial t} |^2 dx
\, dt\\
&= \lim_{{m'} \to \infty} \int_{t'_m-T}^{t'_m+T} \int | \frac{\partial u}{\partial t} |^2 dx
\, dt=0,
\end{align*}
where the last equality is by the finite energy condition. Hence
$|\frac{\partial v}{\partial t}| = 0$ almost everywhere,
which implies that $v$ is an equilibrium.
To prove that $\lim_{t\to\infty}u(t,x)=v(x)$, we follow a relatively
standard line of reasoning, as outlined in
\cite[Proposition 3.19]{Banyaga}. Suppose on the contrary, that $\lim_{t\to\infty} u(t,x)
\not= v(x)$. (Evidently, $v$ is still an accumulation point of $u$.)
Since we assume that the equilibria are isolated, let $U \subset
C^k(K)$ be a closed set with nonempty interior containing $v$ and no
other equilibria. Since $v$ is not a limit of $u$, we can find an
open neighborhood $V \subset U \subset C^k(K)$ of $v$ and a sequence $\{t''_m\}$
with $t''_m \to \infty$ such that $u(t''_m) \in U-V$ for all $m$.
However, the same argument as given in the previous section of the
proof implies that there is an accumulation point $v'$ of
$\{u(t''_m)\}$, and that $v' \in U -V$. We must conclude that $v'$ is
an equilibrium in $U$, which is a contradiction.
Similar reasoning works for
$t\to - \infty$ as we simply then take $t_m \to -\infty$ in the
definition of $u_m$.
\end{proof}
It is not terribly restrictive to assume that the equilibria be
isolated. In one spatial dimension, an equilibrium $f$
of \eqref{pde} is isolated in $C^{2,\alpha}(\mathbb{R})$ (for
$\alpha>0$) when the Schr\"odinger operator $(\frac{d^2}{dx^2} -
2f)$ is injective. This follows from the finiteness of the
point spectrum, which is a consequence of Sturm-Liuoville theory.
Observe that in particular, the injectivity of the Schr\"odinger
operator is therefore generic.
Indeed, we have the following concrete result which ensures that there are
plentiful choices of PDE like \eqref{pde} in which equilibria are
isolated.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\caption{Lower bound for equilibrium $f$ in Lemma \ref{injective_schroedinger}}
\label{f_bnd_fig}
\end{center}
\end{figure}
\begin{lemma}\label{injective_schroedinger}
Suppose that $f$ is an equilibrium of \eqref{pde} which has the form
\[
f(x) \ge \begin{cases}
f_1(x) & \text{for } x < -A\\
f_2(x) & \text{for } x > A\\
-B & \text{for } -A \le x \le A\\
\end{cases}
\]
where $f_1,f_2 > 0$, $f_1' > 0$, $f_2' < 0$ are all continuous, and
$A,B \ge 0$. (See Figure \ref{f_bnd_fig}.) Then for sufficiently small, but nonzero $A,B$, the
Schr\"odinger operator $H=(\frac{d^2}{d x^2} - 2f)$
is injective on the subspace of $C^2(\mathbb{R})$ which consists
of functions that decay to zero.
\end{lemma}
\begin{proof}
Suppose that $Hu=0$ for some nonzero $u \in C^2(\mathbb{R})$ and that
$u(x)$ tends to zero as $|x|\to \infty$. The maximum principle
applied to $\frac{d^2u}{dx^2}=2fu$ implies that $u$ is of one sign on
$x < -A$. Without loss of generality, we assume $u$ is positive.
Indeed, $u$ will be monotonic increasing.
Now suppose that $u'(-A)=u'_0>0$. We solve for a $v$, lower bound on
$u$ defined by
\begin{gather*}
\frac{d^2v}{dx^2} = -2Bv \quad \text{on } -A \le x \le A,\\
v(-A)=u_0 > 0\\
v'(-A)=u'_0 > 0
\end{gather*}
noting that we should require $v''(-A)>0$ by the fact that $u \in
C^2(\mathbb{R})$. Of course, this has the general solution $v=c_1
\cos x\sqrt{2B} + c_2 \sin x\sqrt{2B}$. So if $2A\sqrt{2B}< \pi$,
there can be at most one inflection point of $u$ in $-A \le x \le A$.
By the continuity of $u''$, this means that $u''(+A)>0$. As a result,
the maximum principle implies that $u$ is positive on all of
$\mathbb{R}$. On the other hand, since $f_2>0$, $u(x)$ cannot tend to
zero as $x \to +\infty$, a contradiction.
\end{proof}
We would like to relax the bounds on $u$ and its derivatives, by
showing that they are consequences of the finite energy condition.
The following proposition implies Theorem \ref{limits_to_equilibria}
immediately.
\begin{proposition} \label{finite_energy_consequences_lem}
Suppose that the equilibria of \eqref{pde} are all isolated, and that
either $n=1$ (one spatial dimension) or $N$ is odd. If $u$ is a
finite energy solution to \eqref{pde}, then the the limits
$\lim_{t\to\pm\infty} u(t,x)$ exist uniformly on compact subsets, and
additionally,
\begin{itemize}
\item $u$ is bounded,
\item the derivatives $Du$ are bounded,
\item and therefore the limits are continuous equilibrium solutions.
\end{itemize}
\end{proposition}
\begin{proof}
Since
\[
E(u)=\frac{1}{2}\int_{-\infty}^\infty \int |
\frac{\partial u}{\partial t}
|^2 + | \Delta u + P(u) |^2 dx\, dt < \infty,
\]
we have that for any $\epsilon>0$,
\begin{equation}
\label{lim_infty_eqn}
\lim_{T\to\infty}\frac{1}{2}\int_{T-\epsilon}^{T+\epsilon}
\int | \frac{\partial u}{\partial t}
|^2 + | \Delta u + P(u) |^2 dx\, dt = 0,
\end{equation}
whence $\lim_{t\to\infty} |\frac{\partial u}{\partial t}| =
0$ for almost all $x$. This gives that the limit is an equilibrium
almost everywhere. Of course, this argument works for $t\to -\infty$.
When $N$ is odd, a comparison principle shows that solutions to
\eqref{pde} are always bounded. Observe that for large $|u|$, the
$-u^N$ term dominates the other terms in $P(u)$, which imposes a kind
of asymptotic convexity on the problem.
We need to consider the case with $N$ even. In that case, a
comparison principle on \eqref{pde} shows that $u$ is bounded from
\emph{above}: assume that for a fixed $t_0$, $u(t_0,\cdot)$ attains a
maximum at $x_0$, then
%
\begin{align*}
\frac{\partial u(t_0,x_0)}{\partial t}
&= \Delta u(t_0,x_0) - u^N(t_0,x_0) + \sum_{i=0}^{N-1}
a_i(x_0) u^i(t_0,x_0)\\
&\leq - u^N(t_0,x_0) + \sum_{i=0}^{N-1}
a_i(x_0) u^i(t_0,x_0).
\end{align*}
%
If we assume that $u$ is not bounded from above, then the $u^N$ term
will eventually dominate (since all of the $a_i$ are bounded)
resulting in a contradiction.
On the other hand, if $N$ is even we have assumed that $n=1$ in this
case, and it follows from an asymptotic ODE argument that unbounded
equilibria are bounded from \emph{below}. In one spatial dimension,
equilibrium solutions must satisfy
%
\begin{equation}
\label{equilib_eqn}
u''=u^N-\sum_{i=0}^{N-1}a_i u^i.
\end{equation}
Observe that for $|u|$ sufficiently large, the $u^N$ term will
dominate, since all the $a_i$ are bounded. Therefore, for $|u|$
large, \eqref{equilib_eqn} we have
%
\[
a u^N \le u'' \le A u^N,
\]
%
for some $a,A>0$ whose solutions (when $|u|$ is large) are easily
found (explicitly) to each have a lower bound.
As a result, we must conclude that if a solution to \eqref{pde} tends
to any equilibrium, that equilibrium (and hence $u$ also) must be
bounded.
Now observe that $|\frac{\partial u}{\partial t}|\to 0$ as
$t\to\pm\infty$ on almost all of any compact $K \subset \mathbb{R}^n$ (by
\eqref{lim_infty_eqn}), and that $|\frac{\partial u}{\partial
t}| \le a < \infty$ for some finite $a$ on $\{(t,x)|t=0,x\in
K\}$ by the smoothness of $u$. By the compactness of $K$, this means
that if $\|\frac{\partial u}{\partial
t}\|_{L^\infty((-\infty,\infty)\times K)} = \infty$, there
must be a $(t^*,x^*)$ such that $\lim_{(t,x)\to
(t^*,x^*)}|\frac{\partial u}{\partial t}| = \infty$.
This contradicts smoothness of $u$, so we conclude
$|\frac{\partial u}{\partial t}|$ is bounded on the strip
$(-\infty,\infty)\times K$. On the other hand, the finite energy
condition also implies that for each $v\in\mathbb{R}^n$,
\[
\lim_{s\to\infty} \int_{-\infty}^\infty \int_{K+sv}
|\frac{\partial u}{\partial t} |^2 dx\, dt = 0,
\]
whence we must conclude that $\lim_{s\to\infty}| \frac{\partial
u(t,x+sv)} {\partial t}|=0$ for almost every
$t\in\mathbb{R}$ and $x\in K$. Thus the smoothness of $u$ implies
that $| \frac{\partial u}{\partial t}|$ is bounded on all
of $\mathbb{R}^{n+1}$.
Next, note that since $|\frac{\partial u}{\partial t}|$ and
$u$ are both bounded, then so is $\Delta u$: by \eqref{pde}
\[
\|\Delta u \|_\infty \le \|\frac{\partial u}{\partial
t}\|_\infty + \|u\|_\infty^N + \sum_{i=0}^{N-1} \|a_i\|_\infty
\|u\|_\infty^i,
\]
which uses the boundedness of the $a_i$. Taken together, this implies that all the
spatial first derivatives of $u$ are also bounded.
As a result, we have on $K$ a bounded equicontinuous family of
functions, so Ascoli's theorem implies that they (after extracting a
suitable subsequence) converge uniformly on compact subsets of $K$ to
a continuous limit. As in the end of the proof of Lemma
\ref{parabolic_bootstrap_lem}, the existence of $\lim_{t\to\infty}
u(t,x)$ relies on the equilibria being isolated.
\end{proof}
\subsection{Discussion}\label{disc_sec}
The point of employing the bootstrapping argument of Lemma
\ref{parabolic_bootstrap_lem} is only to extract uniform convergence
of the first derivatives of the solution. As can be seen from the
proof of Proposition \ref{finite_energy_consequences_lem}, such regularity
arguments are unneeded to obtain good convergence of the solution
only.
While Theorem \ref{limits_to_equilibria} is probably true for
all spatial dimensions, the proof given here cannot be generalized to
higher dimensions. In particular, V\'{e}ron in \cite{Veron_1996}
shows that in the case of $P(u)=-u^N$, there are solutions to the
equilibrium equation $\Delta u - u^N=0$ which are \emph{unbounded
below} and \emph{bounded above} when the spatial dimension is greater
than one. This breaks the proof of Proposition
\ref{finite_energy_consequences_lem}, that the limiting equilibria of
finite energy solutions are bounded for $N$ even, since the proof
requires exactly the opposite.
On the other hand, the case of $P(u)=-u|u|^{N-1}+ \sum_{i=0}^{N-1}a_i
u^i$ is considerably easier than what we have considered here. In
particular, all solutions to \eqref{pde} are then bounded. In that
case, the proof of Theorem \ref{limits_to_equilibria} works
for all spatial dimensions.
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