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\AtBeginDocument{{\noindent\small
2005-Oujda International Conference on Nonlinear Analysis.
\newline {\em Electronic Journal of Differential Equations},
Conference 14, 2006, pp. 119--124.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
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\setcounter{page}{119}
\begin{document}
\title[\hfilneg EJDE/Conf/14 \hfil Fast and heteroclinic solutions]
{Fast and heteroclinic solutions for a second order ODE}
\author[M. Arias \hfil EJDE/Conf/14 \hfilneg]
{Margarita Arias}
\address{Margarita Arias \newline
Departamento de Matem\'atica Aplicada \\
Universidad de Granada \\
18071 Granada, Spain}
\email{marias@goliat.ugr.es}
\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{34C37, 35K57, 49J35}
\keywords{Fisher-Kolmogorov's equation; travelling wave solutions;
\hfill\break\indent
speed of propagation; variational methods; constrained minimum problem}
\begin{abstract}
We present some results on the existence of fast and heteroclinic
solutions of an ODE connected with travelling wave solutions of a
Fisher-Kolmogorov's equation. In particular, we present a variational
characterization of the minimum speed of propagation.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
Some chemical and biological systems can be modelled by an
autocatalytic process (see, e.g. \cite{ml,mu}). In many
of these process the system can support propagating wavefronts due
to a combination of a reaction effect and a molecular diffusion.
The pioneering model in this framework is due to Fisher, \cite{f},
who suggested the equation
$$
u_t=u_{xx}+u(1-u)
$$
for studying the spatial spread of a favoured gene in a
population. The simplest generalization of that equation is the so
called Fisher-Kolmogorov's equation
\begin{equation}
u_t=u_{xx}+f(u), \label{1}
\end{equation}
where $f$ is a given function with two zeroes, say $u=0$ and
$u=1$, and positive on $]0,1[$ so that $u=0$ and $u=1$ are the
only two stationary states of (\ref{1}). Equations like (\ref{1})
arises in many problems suggested, for instance, by the classical
theory of population genetics or by certain flame propagation
problems in chemical reactor theory (see, e.g. \cite{aw1}).
A \textit{travelling wavefront} or \textit{travelling wave
solution} (t.w.s., in short) of (\ref{1}) is a solution $u(t,x)$
having a constant profile, that is, such that
$$
u(t,x)=\varphi(x-ct),
$$
for some fixed $\varphi(\xi)$ (called \textit{the wave shape}) and
a constant $c$ (called \textit{the wave speed}). Specially
important for the applications are t.w.s. connecting the two
stationary states, $u=0$ and $u=1$.
A simple calculation shows that if $ u(t,x)=\varphi(x-ct) $ is a
t.w.s. of (\ref{1}), then the wave shape $\varphi$ is a solution
of the ODE
\begin{equation}
u''+cu'+f(u)=0. \label{2}
\end{equation}
When a t.w.s. connects the stationary states, its corresponding
wave shape is a positive heteroclinic solution of (\ref{2}) that
connects the equilibria 1 and 0, that is, a solution of (\ref{2})
defined on $\mathbb{R}$ and satisfying
$$
u(t) \in ]0,1[, \quad \forall t \in \mathbb{R}, \quad
\lim_{t\to
-\infty} u(t)=1, \quad \lim_{t\to +\infty}u(t)=0.
$$
There is a vast and rich body of literature dealing with the existence of
t.w.s. of (\ref{1}) connecting the stationary states, going from
the pioneering work of Kolmogorov, Petrovski and Piskounoff
\cite{kpp}, through the remarkable paper of Aronson and Weinberger
\cite{aw} up to more recent approaches (see \cite{al,m,m',s}).
It is well known (see, e.g. \cite{aw,m}) that there exists a
positive number, $c^*$, such that equation (\ref{2}) has a
heteroclinic solution connecting 1 and 0 if and only if $c\geq
c^*$.
In terms of the Fisher-Kolmogorov's equation, that result says
that none t.w.s. of (\ref{1}) starting from the stationary state
$u=1$ and moving with speed less than $c^*$ reaches the stationary
state $u=0$. $c^*$ is called the \textit{minimum propagation
speed}.
It is clear that the heteroclinic solution, if there exists, is
strictly decreasing. When $f$ is differentiable in $[0,1]$, then
$c^*\geq 2\sqrt{f'(0)}$ since otherwise the origin cannot acts as
an attractor for positive solutions of equation (\ref{2}). It is
also proved (see \cite{aw,m,al,s}) that
$$
c^*\leq 2\sqrt{\sup_{0__0$ approaching
$(0,0)$ has slope at the origin
$$
\lambda_2= \frac{-c+\sqrt{c^2-4f'(0)}}{2}.
$$
Moreover, $T_c$ is extremal in the sense that trajectories below
it stays bounded away from the origin. Aronson and Weinberger (see
\cite{aw}, theorem 4.1) proved that whenever $c^*>2\sqrt{f'(0)}$,
the extremal trajectory $T_{c^*}$ is an heteroclinic solution
between 1 and 0.
This note is a brief summary of the conference given by the author
on the "Colloque International d'Analyse Non lin\'eaire d'Oujda",
about some recent results obtained in collaboration with J.
Campos, A.M. Robles-P\'erez and L. Sanchez
dealing with some variational problems whose
solutions are in correspondence with $T_c$ and that, in
particular, let us give a variational characterization of $c^*$.
All the presented results with their proofs can be found in
\cite{cv}.
\section{A variational characterization of fast solutions}
We say that a solution $u(t)$
of equation (\ref{2}) is \textit{a fast solution} if its
corresponding trajectory is the extremal trajectory $T_c$. Our
purpose is to characterize these solutions in variational terms.
In order to do that, we express their speed in approaching 0 by
means of an integrability condition:
Given $c>0$, we define the space
$$ H_c := \{ u \in H^1_{\rm loc}(0,+\infty) : \int_0^{+\infty}
e^{ct}u'(t)^2\,dt < +\infty \mbox{ and } u(+\infty)=0 \} $$
with the norm $\|u\| = \big(\int_0^{+\infty} e^{ct}u'(t)^2\,dt
\big)^{1/2}$.
This is a Hilbert space and if $u\in H_c$, $u$ obviously tends
``quickly'' to 0 as $t\to + \infty$.
We introduce the functional $\mathcal{F}:H_c\to\mathbb{R}$ defined as
$$\mathcal{F}(u)=\int_0^{+\infty} e^{ct}(\frac{u'(t)^2}{2}-F(u(t)))\,dt, \quad u\in H_c,$$
where $F(u):=\int_0^uf(s)\,ds$. When
\begin{itemize}
\item[(H)]
$f:[0,1]\to\mathbb{R}_+$ is a Lipschitz function such that
$f(0)=0=f(1)$ and $f(u)>0$ if $0____0$, and $u'(t)<0$, for all $t\geq0$, and that $\mathcal{F}$ has a
minimum in $\{u\in H_c:u(0)=1\}$ provided that there exist
$0< k <\frac{c^2}{4}$ with $F(u)\leq ku^2/2$, for all $u \in [0, 1]$.
Therefore, we have the following result.
\begin{proposition} \label{prop1}
Assume (H) and there exist $0< k <\frac{c^2}{4}$ so that
$F(u)\leq ku^2/2$, for all $u \in [0, 1]$. Then equation
\eqref{2} has a fast solution $u\in H_c$ defined on $t\geq0$ such
that $u(0)=1$ and $u'(t)<0$, for all $t\geq0$.
\end{proposition}
This result is particularly connected to the existence of
heteroclinic solutions. Indeed, one can prove that
\begin{quote}
\textit{If there exists a solution of (\ref{2}) defined on $[0,
+\infty)$, with $u(0)=1, \; u(t)>0, \; t>0$ and $u(t)\to 0$ as
$t\to +\infty$, then equation (\ref{2}) has an heteroclinic
solution.}
\end{quote}
So, the above proposition proves the existence of heteroclinic
solutions whenever $\frac{2F(u)}{u^2}\leq \frac{c^2}{4}$, for all
$u\in [0, 1]$. Consequently,
$$
c^* \leq \inf \{ c>0: \frac{2F(u)}{u^2}\leq \frac{c^2}{4}, \quad
\forall u\in [0, 1] \}.
$$
This upper bound generalizes the estimate in \cite{aw}.
\section{Fast heteroclinic solutions}
After studying the fast solutions, we ask about heteroclinic
connections between the two equilibria $u=1$ and $u=0$ of equation
(\ref{2}). As in the previous section, we begin by introducing an
appropriate space to work.
Given $c>0$, we consider the space
$$
X_c:= \{ u \in H^1_{loc}(\mathbb{R}):
\int_{_{-\infty}}^{^{+\infty}} e^{ct}u'(t)^2\,dt < +\infty \mbox{
and } u(+\infty)=0 \},
$$
with the norm $\|u\|_c := \big( \int_{_{-\infty}}^{^{+\infty}}
e^{ct}u'(t)^2\,dt\big)^{1/2}$.
We will say that a solution $u$, of the equation
\begin{equation}\label{eq41}
u''+cu'+\lambda f(u)=0,
\end{equation}
for some $\lambda>0$, is a \textit{fast heteroclinic solution} if
$u\in X_c$ and $u(-\infty)=1$. Note that, under assumption
(H), any heteroclinic connection $u(t)$ of (\ref{eq41}) between
1 and 0 has the property $u(t)\in ]0,1[$, $u'(t)<0$, for all
$t\in\mathbb{R}$.
Our aim now is to obtain a
variational characterization of the smallest value of $\lambda$
for which equation (\ref{eq41}) has a fast heteroclinic solution.
We remark that $u(t)$ is a solution of (\ref{eq41}) for some
$\lambda >0$ if and only if $v(t):=u(t/\sqrt{\lambda})$ is a solution of
(\ref{2}) with $c= c/\sqrt{\lambda}$.
To do that, we introduce two real functionals on $X_c$:
$$
A_c(u) := \int_{-\infty}^{+\infty} e^{ct} \frac{u'(t)^2}{2}\,dt; \quad
B_c(u) := \int_{-\infty}^{+\infty} e^{ct}F(u(t))\,dt,
$$
and we will look for critical points of the restriction of $A_c$ to the
set $M_c:=\{ u\in X_c: B_c(u)=1 \}$. (Note that $M_c$ is non
empty as a consequence of the hypothesis on $f$).
We define
$$
\lambda(c) :=\inf\{A_c(u): u\in M_c \}.
$$
It is easy to check that $A_c$ and $B_c$ are $C^1$-functionals and
$M_c$ is a $C^1$-manifold. By Lagrange multipliers rule, $u\in
M_c$ is a critical point of the restriction of $A_c$ to $M_c$ if
and only if $u\in M_c$ is a solution of (\ref{eq41}). Playing
appropriately with (\ref{eq41}) we are able to prove that
\begin{quote}
\textit{If $\lambda(c)$ is attained, then equation (\ref{eq41})
with $\lambda =\lambda(c)$ has a fast heteroclinic solution $u\in
M_c$ and $A_c(u)=\lambda(c)$.}
\end{quote}
\begin{remark} \label{rmk0} \rm
Given $u\in X_c$ and $a \in \mathbb{R}$, the function $v(t):=u(t-a)$
belongs to $X_c$ and $A_c(v)=e^{ca}A_c(u$),
$B_c(u)=e^{ca}B_c(u)$. So, if $u\in X_c$ is a critical point of
$A_c$ subject to the restriction $B_c(u)=1$, for all $\alpha
>0$, the function $v(t):=u(t-\frac{\ln \alpha}{c})$ is a critical
point of $A_c$ subject to the restriction $B_c(u)=\alpha$. Hence,
condition $B_c(u)=1$ is a kind of normalization.
\end{remark}
The previous result reduces the problem of the existence of fast
heteroclinic solutions to prove that $\lambda(c)$ is attained.
Using a convenient closed convex set, we show that $\lambda(c)$ is
attained when $F(u)=o(u^2)$ as $u\to 0^+$.
Finally, working with an auxiliary functional defined on that
closed convex set, we obtain our main result.
\begin{theorem} \label{teo44}
Assume (H), and also that there exists $f'(0)$ and
\begin{equation}\label{eqH3}
\lambda(c) < \frac{c^2}{4f'(0)}.
\end{equation}
Then, $\lambda(c)$ is attained. In particular, (\ref{3}) with
$\lambda=\lambda(c)$ has a fast heteroclinic solution.
\end{theorem}
Observe that our approach does not require differentiability
except at the origin. On the other hand, if there exists $f'(0)$,
working with truncations of the function $\varepsilon e^{-kt}$,
$\varepsilon \to 0$, $k\downarrow c/2$, one can prove
$$
\lambda(c) \leq \frac{c^2}{4f'(0)},
$$
and condition (\ref{eqH3}) is almost necessary.
Moreover, as a consequence of this result, if there exists
$f'(0)$, $\lambda(c)$ is positive. A simple change of variable
shows that $\lambda(c)=c^2\lambda(1)$. Hence, condition
(\ref{eqH3}) is independent of $c$ and it can be write
$$
\lambda(1) < \frac{1}{4f'(0)}.
$$
\section{A variational characterization of $c^*$}
Theorem \ref{teo44} let us obtain a variational characterization
of the \textit{minimum propagation speed} $c^*$. As we
have already said in the introduction,
$$
c^*:=\inf\{ c\in \mathbb{R}: (\ref{2})
\mbox{ has an heteroclinic solution.} \}
$$
Mallaguti and Marcelli \cite{m'} proved that $c^*$ is in fact a
minimum and it is positive. We are going to relate this number
with the function $\lambda(c)$ introduced
in the previous section. In order to do that, let us define
$$\bar c :=\frac{1}{\sqrt{\lambda(1)}}.$$
Having in mind that $\lambda(c) \leq \frac{c^2}{4f'(0)}$ and
$\lambda(c)=c^2\lambda(1)$, one has that $\bar c \geq
2\sqrt{f'(0)}$.
From Theorem \ref{teo44}, if $\bar c > 2\sqrt{f'(0)}$, equation
(\ref{2}) with $c=\bar c$ has a fast heteroclinic solution and,
then, $\bar c \geq c^*$. We can prove the following result.
\begin{theorem} $\bar{c}=c^*$.
\end{theorem}
The proof of this theorem is based on the following result.
\begin{proposition}\label{prop51}
Assume that for some $c>2\sqrt{f'(0)}$ there exists an
heteroclinic solution. Then, $c=c^*$ if and only if this
heteroclinic is fast.
\end{proposition}
Remark that the previous proposition says:
\begin{quote}
\textit{At least when $c>2\sqrt{f'(0)}$, $c^*$ is the only value
of the parameter for which the heteroclinic connection between the
two equilibria of (\ref{2}) is fast.}
\end{quote}
The proof of this result follows by interpreting positive
decreasing solutions of (\ref{2}) as solutions of a suitable first
order equation (as it has been done in \cite{sa,m}).
A positive decreasing solution of (\ref{2}) has a trajectory in
the second quadrant of the $(u,u')$-plane. It is about looking at
such a trajectory as the graph of a function $\phi$, so that
$u'=\phi(u)$. Putting $y(u)=\phi(u)^2$, $y$ is a solution of
\begin{equation}\label{eq53}
\frac{dy}{du} =2c\sqrt y-2f(u).
\end{equation}
A heteroclinic solution of (\ref{2}) corresponds to a positive
solution of (\ref{eq53}) on $]0,1[$ such that $y(0)=y(1)=0$.
(Note that the Cauchy problem for equation (\ref{eq53}) has no
uniqueness, but any solution of (\ref{eq53}) can be continued as
long as it remains positive.)
Summarizing, we obtain
$$
c^*= \Big( \inf \Big\{ \int_{-\infty}^{+\infty} e^{ct}
\frac{u'(t)^2}{2}\,dt\,: \; u\in X_1, \;
\int_{_{-\infty}}^{^{+\infty}} e^{ct}F(u(t))\,dt =1
\Big\}\Big)^{-1}.
$$
Moreover, when $c>c^*$ equation (\ref{2}) has an heteroclinic
connection between its equilibria though it is no fast, that is,
\textit{the extremal trajectory $T_c$ is not an heteroclinic}, but
if $c=c^*>2\sqrt{f'(0)}$, then $T_c$ connects the two
equilibria.
When $c^*=2\sqrt{f'(0)}$, (\ref{2}) has an heteroclinic connection
between its equilibria, but it is an open problem to know if it is
or not a fast heteroclinic.
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\end{document}
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