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\title[Linear vs. nonlinear selection]{Linear vs. nonlinear selection
for the propagation speed of the solutions of scalar
reaction-diffusion equations invading an unstable equilibrium}

\author[M. Lucia, C. B. Muratov, and M. Novaga]{}

\subjclass{35K57}

\email{muratov@njit.edu}

\keywords{Reaction-diffusion systems, Traveling waves, Long-time
asymptotics, Wave selection}

\address{Department of Mathematical Sciences, New Jersey Institute of
Technology, Newark, NJ 07102}

\begin{document}

\maketitle

\centerline{\scshape M. Lucia} \medskip

  {\footnotesize \centerline{ Department of Mathematics, Rutgers
  University } \centerline{ 110 Frelinghuysen Rd, Piscataway NJ 08854,
  USA} } \medskip

\centerline{\scshape   C. B. Muratov}
 \medskip

  {\footnotesize \centerline{ Department of Mathematical Sciences }
  \centerline{ New Jersey Institute of Technology, Newark, NJ 07102, USA }
  } \medskip

\centerline{\scshape   M. Novaga}
 \medskip

  {\footnotesize \centerline{ Dipartimento di Matematica, Universit\`a
  di Pisa } \centerline{ Via Buonarotti 2, Pisa 56127, Italy} }
  \medskip

\begin{abstract}

We revisit the classical problem of speed selection for the
propagation of disturbances in scalar reaction-diffusion equations
with one linearly stable and one linearly unstable equilibrium. For a
wide class of initial data this problem reduces to finding the minimal
speed of the monotone traveling wave solutions connecting these two
equilibria in one space dimension. We introduce a variational
characterization of these traveling wave solutions and give a
necessary and sufficient condition for linear vs. nonlinear selection
mechanism. Easily verifiable sufficient conditions for the linear and
nonlinear selection mechanisms are obtained. Our method also allows to
obtain efficient lower and upper bounds for the propagation speed.

\end{abstract}

\section{Introduction}

In this paper, we revisit the problem of speed selection for the
propagation of disturbances in scalar reaction-diffusion systems. To
this end, we consider the following initial value problem:
\begin{eqnarray}
\label{rd}
u_t = u_{xx} + f(u), \hspace{1cm} u(x, 0) = u_0(x),
\end{eqnarray}
where $u = u(x, t)$ is a real-valued function of one spatial variable
$x$ and time $t$. For simplicity, we consider the problem on a real
line, a straightforward generalization to the cylindrical geometry
with Neumann boundary conditions or the entire $\mathbb R^n$ is
possible. Furthermore, we will consider nonlinearities $f$ possessing
an unstable equilibrium (which without any loss of generality may be
assumed to be zero) and a stable equilibrium of the space-independent
dynamics governed by Eq.~(\ref{rd}), with no other equilibria in
between.

This kind of equation is a prototypical model for a variety of
applications in physics, chemistry, and biology (see, for example,
\cite{cross93,merzhanov99,field,murray}). One of the most notable
examples is the Fisher equation with nonlinearity $f(u) = u(1 - u)$
describing the spread of advantages genes in a population, whose
studies go back to 1930's
\cite{murray,fisher37,kolmogorov37}. Equation (\ref{rd}) arises
naturally in the context of autocatalytic reactions and combustion
systems, for example, in the case of the Arrhenius nonlinearity one
gets Eq.~(\ref{rd}) with $f(u) = e^{-{a \over u}}(1 - u)$ with $a > 0$
\cite{cross93,merzhanov99,field}. Another important class of problems
which leads to Eq.~(\ref{rd}) with an unstable equilibrium arises from
the analysis of amplitude equations describing the dynamics of the
system near a bifurcation point \cite{cross93}. As one such
characteristic example, consider the subcritical quintic
Ginzburg-Landau equation
\begin{eqnarray}
\label{quint}
u_t = u_{xx} + \mu u + u^3 - u^5,
\end{eqnarray}
which has an unstable equilibrium $u = 0$ for $\mu > 0$.

We are interested in the process of invasion of the unstable
equilibrium by a stable one in reaction-diffusion systems described by
Eq.~(\ref{rd}). So, we will consider the solutions of Eq.~(\ref{rd})
with the initial data decaying exponentially as $x \to +\infty$. These
solutions are known to exhibit propagation with constant speed at long
times. Back in the 30's, Kolmogorov, Petrovsky, and Piskunov showed
that Eq.~(\ref{rd}) admits a particular class of solutions in the form
of {\em traveling waves}, $u(x, t) = \bar u(x - c t)$, moving with
speed $c$, where the profile of the wave satisfies the ordinary
differential equation
\begin{eqnarray}
\label{trav}
\bar u_{xx} + c \bar u_x + f(\bar u) = 0.
\end{eqnarray}
Under some extra assumptions on $f$, they were able to prove that
there exists a continuous family of the traveling wave solutions with
arbitrary speeds $c \geq c_0$, which decay exponentially with the rate
depending on $c$ \cite{kolmogorov37}. This fact already indicates that
the speed of propagation for the solutions of the considered initial
value problem may depend on the way the initial data go to zero at
plus infinity. However, as was shown already by Kolmogorov, Petrovsky,
and Piskunov, for the initial data that decay sufficiently rapidly, in
particular, if $u_0(x) \equiv 0$ for all $x > x_0$, for such
nonlinearities the propagation speed turns out to be equal to $c_0$,
which, furthermore, is easily calculated from the linearization of
Eq.~(\ref{rd}) around zero.

The problem of speed selection was discussed extensively in the
physics literature (see, for example,
\cite{benjacob85,vansaarloos88,vansaarloos89,vansaarloos03,%
powell91,benguria96}). It was observed that depending on the
nonlinearity, the propagation speed $c^*$ for sufficiently rapidly
decaying initial conditions is either equal, or is greater than the
speed $c_0$ obtained by Kolmogorov, Petrovsky, and Piskunov. Depending
on whether the first or the second situation is realized, the
selection mechanism is termed ``linear'' or ``nonlinear'' selection,
respectively. Van Saarloos argued that whether the first or the second
mechanism is realized depends on the existence of certain types of the
traveling wave solutions
\cite{vansaarloos88,vansaarloos89,vansaarloos03}. Rigorous studies of
this problem in the general context of Eqs.~(\ref{rd}) were initiated
by Aronson and Weinberger \cite{aronson75,aronson78}. They essentially
resolved the problem of the speed selection with the help of the
comparison arguments for a certain class of initial data. Central to
their analysis is the construction of the monotone traveling wave
solutions connecting the stable equilibrium at minus infinity with the
unstable equilibrium at plus infinity. Aronson and Weinberger proved
that under certain assumptions these solutions in fact exist for all
$c \geq c^*$, where $c^*$ is some constant which may be equal to or
strictly greater than $c_0$. Furthermore, they proved that any
nontrivial solution of the initial value problem in Eq.~(\ref{rd})
with the initial data $u_0(x)$ between these equilibria and decaying
exponentially, faster than the traveling wave solution with speed
$c^*$, will propagate asymptotically with speed $c^*$ in the sense
that the position $R(t)$ of the {\em leading edge} of the solution of
Eq.~(\ref{rd}) behaves asymptotically as (for precise definitions and
assumptions, see Section \ref{s:prel})
\begin{eqnarray}
\label{R}
R(t) \sim c^* t, \hspace{1cm} t \to +\infty.
\end{eqnarray}

More recently, Roquejoffre proved that when $c^* > c_0$ and under some
mild assumptions any solution of the initial value problem in
Eq.~(\ref{rd}), with the initial data decaying sufficiently fast at
plus infinity and bounded away from zero at minus infinity converges
exponentially to the traveling wave solution with speed $c^*$
\cite{roquejoffre97}. Thus, he proved not only that under these
assumptions the position of the leading edge behaves asymptotically
according to Eq.~(\ref{R}), but also that the wave profile approaches
uniformly a traveling wave profile with speed $c^*$ (this result
remains valid in cylinders with Neumann boundary conditions, and even
in the presence of advection terms). Convergence results for this type
of equations were also recently discussed from a variational
perspective \cite{m}.

 From the discussion above it is clear that under rather general
assumptions on the initial data the problem of speed selection reduces
to finding the minimal speed of the monotone traveling wave solutions
connecting the two equilibria of $f$. While Aronson and Weinberger
give a definitive answer about the existence of the propagation speed
$c^*$, that is, the existence of the limit in Eq.~(\ref{R}), their
techniques do not say when the value of $c^* = c_0$ and when $c^* >
c_0$. In other words, they do not provide a way to establish whether
the linear or nonlinear selection mechanism is realized. Note that
this is also a necessary ingredient for applying the results of
\cite{roquejoffre97}. Therefore, the problem of characterizing the
value of $c^*$ is of fundamental importance for understanding the
long-time behavior of the solutions of Eq.~(\ref{rd}).

In this paper, we develop a variational characterization of the
traveling wave solutions with speed $c^* > c_0$. We give a necessary
and sufficient condition for existence of this solution and thus give
a verifiable answer to the question of linear vs. nonlinear
selection. Our method also allows to obtain efficient upper and lower
bounds for $c^*$ which can be easily implemented in practice.

Our paper is organized as follows. In section \ref{s:prel}, we
introduce some notation and summarize known results about propagation
for solutions of Eq.~(\ref{rd}). We also generalize the main results
of Aronson and Weinberger for the limiting behavior of the propagation
speed to initial data with exponential decay and state our main
result. In section \ref{s:constr}, we introduce our variational
problem and demonstrate its relationship to the existence of some
special traveling wave solutions which determine the limiting
propagation speed. In section \ref{s:exist} we prove our main Theorem,
in section \ref{s:appl} we consider a few applications of our results,
and in section \ref{s:concl} we make concluding remarks.

\section{Preliminaries and the main result} \label{s:prel}

Here we give a few basic definitions and state our main result. Let us
start by giving some known results that will be used throughout the
paper. We look for the solutions of the initial value problem in
Eq.~(\ref{rd})
\begin{eqnarray}
u(x, t): \mathbb R \times \mathbb R^+ \to \mathbb R,
\mathrm{~~~with~~~} u_0(x) \in C(\mathbb R).
\end{eqnarray}
We assume that the nonlinearity $f \in C^1([0, 1])$ and
satisfies
\begin{eqnarray}
\label{ass}
f(0) = f(1) = 0, \hspace{1cm} f'(0) > 0 > f'(1), \hspace{1cm} f(u) > 0
\mathrm{~for~} 0 < u < 1.
\end{eqnarray}
Thus, $u = 0$ is the unstable, and $u = 1$ is the stable equilibrium.
We will further consider the initial data bounded to the strip
\begin{eqnarray}
\label{b01}
0 \leq u_0(x) \leq 1.
\end{eqnarray}
By maximum principle, the solutions of the initial value problem in
Eq.~(\ref{rd}) with these initial conditions will remain bounded
between zero and one and therefore will exist for all $t > 0$ (see,
for example, \cite{fife}).

Turn now to the traveling wave solutions. First, any bounded solution
of Eq.~(\ref{trav}) must connect the equilibria at infinity. In
particular, with our assumptions on $f$ we have
\begin{proposition}[Volpert {\em et al.} \cite{volpert}]
\label{p:volpert}
Let $\bar u(x) \in [0, 1]$ be a solution of Eq.~(\ref{trav}) with $c
> 0$. Then
\begin{eqnarray}
\lim_{x \to +\infty} \bar u(x) = 0, \label{ubpi} \\ \lim_{x \to
-\infty} \bar u(x) = 1 \label{ubmi},
\end{eqnarray}
and, furthermore, $\bar u(x)$ is monotonically decreasing.
\end{proposition}
Naturally, because of reflection symmetry for any solution of
Eq.~(\ref{trav}) there exists a mirror-symmetric solution of this
equation with speed $-c$ (hence, with the reversed order of the limits
in Eqs.~(\ref{ubpi}) and (\ref{ubmi})). Therefore, in the following we
will only consider the right-moving waves and propagation toward $x =
+\infty$. 

Let us define a positive constant 
\begin{eqnarray}
\label{c0}
c_0 = 2 \sqrt{f'(0)}.
\end{eqnarray}
It is easy to see that this constant plays a role of the minimal
possible propagation speed for monotone traveling wave
solutions. Indeed, the behavior ahead of the wave is determined by the
linearization of Eq.~(\ref{trav}) around zero. Assuming that $\bar u
(x) \sim e^{-\lambda x}$, we obtain an equation 
\begin{eqnarray}
\lambda^2 - c \lambda + f'(0) = 0,
\end{eqnarray}
relating $c$ and $\lambda$, whose solutions are $\lambda =
\lambda_\pm(c)$, where
\begin{eqnarray}
\lambda_\pm(c) = \frac{1}{2} \left( c \pm \sqrt{c^2 - 4 f'(0)}
\right).
\end{eqnarray}
 From this equation follows that the decay of $\bar u(x)$ for $x \to
+\infty$ is non-oscillatory only if $c \geq c_0$.

The existence of the traveling wave solutions that satisfy the
conditions in Eqs.~(\ref{ubpi}) and (\ref{ubmi}) for the
nonlinearities satisfying Eq.~(\ref{ass}) were proved by Aronson and
Weinberger:
\begin{proposition}[Aronson and Weinberger \cite{aronson75,aronson78}]
\label{awtrav} 
Under the assumptions of Eq.~(\ref{ass}), for each $c \geq c^*$, with
some $c^* \geq c_0$, there exists a unique (up to translations)
solution $\bar u(x)$ of Eq.~(\ref{trav}) satisfying Eqs.~(\ref{ubpi})
and (\ref{ubmi}). Furthermore, for $c > c^*$ we have
\begin{eqnarray}
\label{lam-}
\ln \bar u(x) \sim -\lambda_-(c) x, \hspace{1cm} x \to +\infty,
\end{eqnarray}
whereas for $c = c^*$ we have
\begin{eqnarray}
\label{lam+}
\ln \bar u(x) \sim -\lambda_+(c^*) x, \hspace{1cm} x \to +\infty.
\end{eqnarray}
When $c < c^*$, there are no traveling wave solutions satisfying the
assumption of Proposition \ref{p:volpert}.
\end{proposition}

Let us now define the position of the leading edge $R(t)$ of the
solution of the initial value problem in Eq.~(\ref{rd}). Let $\alpha
\in (0, 1)$ be some fixed constant which is so small that $\max_{x \in
\mathbb R}u(x, t) \geq \alpha$ for $t \geq 0$. Clearly, such a
constant necessarily exists at least for large time intervals, if the
initial data are not zero identically. Denote
\begin{eqnarray}
R(t) = \sup \{ x \in \mathbb R: u(x, t) \geq \alpha \}.
\end{eqnarray}
In terms of $R(t)$, we have the following Theorem, which is a
generalization of the classical results of Aronson and Weinberger. 

\begin{theorem}
\label{awspeed}
Let $u(x, t)$ be a solution of Eq.~(\ref{rd}) with the initial
condition not equal to zero identically, satisfying Eq.~(\ref{b01})
and
\begin{eqnarray}
\label{linf}
\limsup_{x \to +\infty} u_0(x) e^{\lambda x} < +\infty,
\end{eqnarray}
where $\lambda = \lambda_-(c^*)$. Then there exists a constant
$\alpha$ such that Eq.~(\ref{R}) holds, with $c^*$ the same as in
Proposition \ref{awtrav}.
\end{theorem}

\begin{proof}
We first note that $R(t)$ is well-defined for all $t \geq 0$, if
$\alpha$ is small enough. Indeed, by the results of Aronson and
Weinberger, for any non-zero initial condition we have $\lim_{t \to
+\infty} \max_{x \in \mathbb R} u(x, t) = 1$ (see
\cite{aronson75,aronson78}), so it is always possible to choose
$\alpha$ such that $\max_{x \in \mathbb R} u(x, t) \geq \alpha$ for
all $t \geq 0$. Of course, $\max _{x \in \mathbb R} u(x, t) > 0$ for
all $t \geq 0$, otherwise by uniqueness of solutions of Eq.~(\ref{rd})
we would have $u(x, t) \equiv 0$.

Let us first show that $R(t) \geq c t$ for any $c < c^*$ and large
enough $t$. This follows immediately from the fact that by the results
of Aronson and Weinberger $u \to 1$ pointwise in any reference frame
moving with speed $0 < c < c^*$ \cite{aronson75,aronson78}. 

To prove that $R(t) \leq ct + R_0$ for all $t$, with arbitrary $c >
c^*$ and some $R_0$, we construct an appropriate supersolution in the
reference frame moving with speed $c$. To do that, let us note that
for $c > c^*$ there is a unique trajectory starting at the origin and
having a slope $-\lambda_+(c)$ in the phase plane
\cite{aronson75,aronson78}. Since $c > c^*$, this trajectory
intersects the line $u = 1$ at some $u_x = \nu_0 < 0$. By uniqueness
of the above trajectory, any other phase plane trajectory going into
the origin must have the slope $-\lambda_-(c)$. On the other hand, by
Proposition \ref{awtrav}, for $c > c^*$ there exists a traveling wave
solution connecting $u = 0$ and $u = 1$, with slope
$-\lambda_-(c)$. Therefore, any phase plane trajectory starting at $u
= 1$ and $u_x = \nu$ with $\nu_0 < \nu < 0$ will terminate at the
origin and will have the slope $-\lambda_-(c)$ as well. We now
construct a supersolution $\bar u(x)$ in the reference frame moving
with speed $c$ by assuming that $\bar u(x)$ satisfies Eq.~(\ref{trav})
for $x \geq 0$ with the initial conditions $\bar u(0) = 1$, $\bar
u_x(0) = \nu$, and take $\bar u = 1$ for $x < 0$.

Now, observe that $\lambda_-(c) < \lambda_-(c^*)$ for $c > c^*$.
Then, if Eq.~(\ref{linf}) holds, we can always bound $u_0$ with a
translate of $\bar u$ from above. Since also $\bar u$ is monotonically
decreasing and $\bar u(x) \to 0$ as $x \to +\infty$, this implies that
$R(t) \leq c t + R_0$ with some $R_0$. Finally, since $c$ can be
arbitrarily close to $c^*$ in both the upper and lower bound for
$R(t)$, we obtain Eq.~(\ref{R}).
\end{proof}

Thus, the asymptotic propagation speed for the solutions of
Eq.~(\ref{rd}) with the initial data satisfying the assumptions of
Theorem \ref{awspeed} is $c^*$ in the sense of the asymptotic average
speed of the leading edge. Note that the speed $c^*$ is also an
asymptotic lower bound for the speed of the leading edge for {\em any}
nontrivial initial condition (see the proof above). Furthermore, note
that the assumption on the decay in Eq.~(\ref{linf}) is almost
necessary, since if $\lambda < \lambda_-(c^*)$, then solutions whose
leading edge moves faster than $c^*$ (like the traveling waves with
speed $c > c^*$ in Proposition \ref{awtrav}) are possible.

\begin{remark} 
We point out that in view of \cite{aronson75} the function $R(t)$ is
defined for any $\alpha \in (0, 1)$ when $t$ is large enough (see also
the proof above), so the statement of Theorem \ref{awspeed} remains
valid for all such $\alpha$. Also note that Eq.~(\ref{R}) holds
independently of $\alpha$.
\end{remark}


We have two possibilities which we need to discern, so, following van
Saarloos \cite{vansaarloos88,vansaarloos89,vansaarloos03}, we
introduce the following

\begin{definition}
Under the assumptions of Theorem \ref{awspeed}, we call the case $c^*
= c_0$ the {\em linear selection}, while the case $c^* > c_0$ the {\em
nonlinear selection} mechanism.
\end{definition}

The purpose of what follows is to characterize linear vs. nonlinear
selection for a given nonlinearity $f(u)$ obeying Eq.~(\ref{ass})
within the context of Theorem \ref{awspeed}. We start by defining the
exponentially weighted Sobolev spaces we will be working in.

\begin{definition}
For $c > 0$, denote by $H^1_c(\mathbb R)$ the completion of
$C_0^\infty(\mathbb R)$ with respect to the norm
\begin{eqnarray*}
  ||u||_{1,c} = ||u||_c + ||u_x||_c, \qquad ||u||^2_c = \int_{\mathbb
    R} e^{cx} u^2 dx.
\end{eqnarray*}
\end{definition}
These are in fact natural spaces for working with the solutions of
Eq.~(\ref{trav}), since they provide control of the exponential decay
of the solution at plus infinity for different choices of the constant
$c$ (see below, and also \cite{m}).

For $u \in H^1_c(\mathbb R)$, define the functional
\begin{eqnarray}
\label{phi}
\Phi_c[u] = \int_\mathbb{R} e^{cx} \left( \frac{1}{2} u_x^2 + V(u)
\right) dx,  
\end{eqnarray}
where the function $V(u)$ is given by
\begin{eqnarray}
V(u) = 
\begin{cases}
\frac{1}{2} |f'(0)| u^2, & u < 0, \\ - \int_0^u f(s) ds, & 0
\leq u \leq 1, \\ - \int_0^1 f(s) ds + \frac{1}{2} |f'(1)|(u - 1)^2, &
u > 1.
\end{cases}
\end{eqnarray}
 From the assumptions on $f(u)$ in Eq.~(\ref{ass}), it immediately
follows that $V(u) \in C^1(\mathbb R)$ and that $|V(u)| \leq C u^2$
for some $C$, so $\Phi_c[u]$ is well-defined for all $u$ in the
considered class. 

We now state our main result. 

\begin{theorem}
\label{main}
Under the assumptions of Theorem \ref{awspeed}, the nonlinear
selection mechanism is realized if and only if 
there exists $u \in H^1_c(\mathbb R)$, $u \not \equiv 0$, such that
\begin{eqnarray} \label{neg}
\Phi_c[u] \leq 0
\end{eqnarray}
for some $c > c_0$.
\end{theorem}

Thus, the functional $\Phi_c$ provides a complete characterization of
the speed selection mechanism within the framework of Aronson and
Weinberger. 

\section{Constrained variational problem} \label{s:constr}

We now formulate the problem of existence of the traveling wave
solutions with speed $c^*$ as a constrained variational problem. We
point out that our method provides a very general way of constructing
the traveling wave solutions and is not limited to the case of the
nonlinearities specified in Eq.~(\ref{ass}), or one-dimensional scalar
reaction diffusion equations. The general treatment of this problem
from the variational perspective will be presented elsewhere
\cite{hlmn}.

For $u \in H^1_c(\mathbb R)$, introduce an auxiliary functional
\begin{eqnarray}
\Gamma_c[u] = \frac{1}{2} \int_\mathbb{R} e^{cx}
u_x^2 \, dx.
\end{eqnarray}
Note that both $\Phi_c$ and $\Gamma_c$ transform similarly under
translations.
\begin{lemma}
\label{lem:Translation}
Let $u \in H^1_c(\mathbb R)$ and $u_a(x) = u(x - a)$. Then, 
\begin{eqnarray}
\Phi_c[u_a] = e^{ca} \Phi_c[u] \quad \mathrm{and} \quad \Gamma_c[u_a]
= e^{ca} \Gamma_c[u].
\end{eqnarray}
\end{lemma}

Now, by setting
\begin{eqnarray}
{\mathcal B} = \{ u \in H^1_c (\mathbb R) : \Gamma_c [u] =1 \},
\end{eqnarray}
we obtain the following constrained minimization problem:
\begin{itemize}
\item[(P)] \quad {\it Find $u_c \in {\mathcal B}$ satisfying:
$\displaystyle \Phi_c [u_c] = \inf_{ {\mathcal B} } \Phi_c [u]$.}
\end{itemize}
The connection between the solutions of problem (P) and the solutions
of Eq.~(\ref{trav}) is established by the following

\begin{proposition} 
\label{travex} Let $u_c(x)$ be a solution of problem (P) with
$\Phi_c[u_c] \leq 0$. Then
\begin{eqnarray}
\bar u(x) = u_c(x \sqrt{1 - \Phi_c[u_c]})
\end{eqnarray}
is the traveling wave solution with speed $c^* = c \sqrt{1 -
  \Phi_c[u_c]}$. Furthermore, $c^*$ is the same as in Proposition
  \ref{awtrav}.
\end{proposition}

We prove this proposition via a sequence of lemmas.

\begin{lemma}
\label{l01}
Let $u_c(x)$ be a solution of problem (P) with $\Phi_c[u_c] \leq
0$. Then $0 \leq u_c(x) \leq 1$.
\end{lemma}

\begin{proof}
We first note that $u \in H^1_c(\mathbb R)$ implies $u \in C(\mathbb
R)$. Observe that for any $u \in H^1_c(\mathbb R)$ we can define
\begin{eqnarray}
\tilde u(x) = 
\begin{cases}
0, & u(x) < 0,\\ u(x), & 0 \leq u(x) \leq 1, \\ 1, & u(x) > 1.
\end{cases}
\end{eqnarray}
Since $V(u)$ is strictly increasing outside the interval $u \in [0,
1]$, we have
\begin{eqnarray}
\label{decr}
\Phi_c[\tilde u] < \Phi_c[u] \leq 0,
\end{eqnarray}
unless $u \in [0, 1]$. We also have $\Gamma_c [\tilde u] \leq
1$. Furthermore, $\Gamma_c[\tilde u] > 0$, since otherwise $\tilde u =
0$ and hence $\Phi_c[\tilde u] = 0$, contradicting
Eq.~(\ref{decr}). Therefore, by Lemma \ref{lem:Translation} it is
always possible to find a value of $a \geq 0$ that $\tilde u_a(x) =
\tilde u(x - a) \in \mathcal B$, too. So, if $u$ is the solution of
problem (P), then so is $\tilde u_a$, which by Lemma
\ref{lem:Translation} and Eq.~(\ref{decr}) gives lower value of
$\Phi_c$, leading to contradiction.
\end{proof}


\begin{lemma}
\label{weak}
Let $u = u_c(x)$ be a solution of problem (P). Then $u$ satisfies
\begin{equation} \label{eq:Lagrange2}
 \int_{\mathbb R} e^{cx} \{ (1 - \mu) u_x \varphi_x + V'(u)
 \varphi \} dx = 0 \qquad \forall \varphi \in H^1_c (\mathbb R).
\end{equation}
Moreover, 
\begin{eqnarray}
\label{lam}
\mu = \Phi_c[u_c].
\end{eqnarray}
\end{lemma}

\begin{proof}
We have $\Phi_c$ and $\Gamma_c$ of class $C^1$. Let $D \Gamma_c[u] v$
be the Frechet derivative of $\Gamma_c$ at $u$ acting on $v$. Since
$$ D \Gamma_c [u] u = \int_{\mathbb R} e^{cx} u_x^2 dx = 2 \quad
  \forall u \in {\mathcal B},
$$ we get $D \Gamma_c [u] \not \equiv 0$ on ${\mathcal B}$. Thus,
applying the Lagrange Multiplier Theorem (see, for example,
\cite[Section 3.5]{LagThm}), we obtain Eq.~(\ref{eq:Lagrange2}), where
$\mu$ is the Lagrange multiplier.

Let us now prove that $\mu = \Phi_c[u]$. First of all, we have
$\mu \not = 1$. Indeed, if $\mu = 1$, from
Eq.~(\ref{eq:Lagrange2}) we get $V'(u) \equiv 0$. Since also $u$ is
continuous, according to Eq.~(\ref{ass}) this means that $u \equiv 0
\not \in \mathcal B$, leading to contradiction.

So, $\mu \not = 1$, and from the elliptic regularity theory we
deduce that $u \in C^2(\mathbb R)$, and furthermore $u_x \in
H^1_c(\mathbb R)$ (see, for example, \cite{gilbarg}). Therefore, we
can use $\varphi = u_x$ in Eq.~(\ref{eq:Lagrange2}). Integrating by
parts, we obtain
\begin{eqnarray*}
0 = \int_\mathbb{R} e^{cx} \{ (1 - \mu) u_x u_{xx} + V'(u) u_x \}
dx \nonumber \\ = - c \int_\mathbb{R} e^{cx} \left( \frac{1}{2} (1 -
\mu) u_x^2 + V(u) \right) = - c (\Phi_c[u] - \mu \Gamma_c[u]).
\end{eqnarray*}
So, taking into account that $u \in \mathcal B$, we obtain
Eq.~(\ref{lam}). 


%% Given $R > 0$, let $\eta \in
%% C_0^{\infty} (\mathbb R)$ be such that:
%% $$ {\rm supp} (\eta) \subset (-R,R), \quad \eta (x) = 1 \quad \forall
%%  x \in (-\frac{R}{2}, \frac{R}{2}) \quad |\eta_x| \leq C,
%% $$ where $C$ does not depend on $R$.  By considering $\varphi = \eta
%% u_x$ as a test function in (\ref{eq:Lagrange2}), we get
%% $$ \int_{\mathbb R} \left\{ (1 - \mu) u_x (\eta_x u_x + \eta
%%   u_{xx}) + V'(u) \eta u_x \right\} e^{cx} = 0
%% $$
%% $$
%%   (1 - \mu) \int_{-R}^{R} |u_x|^2 e^{cx} \eta_x dx
%%   +
%%   \int_{-R}^{R} \left[ 
%%     (1 - \mu) \frac{|u_x|^2}{2} + V(u) 
%%   \right]_x e^{cx} \eta = 0. 
%% $$ 
%% $$
%%   (1 - \mu) \int_{-R}^{R} |u_x|^2 e^{cx} \eta_x dx
%%   -
%%   \int_{-R}^{R} \left\{
%%     (1 - \mu) \frac{|u_x|^2}{2} + V(u)
%%   \right\} (\eta e^{cx})_x dx  
%% $$
%% Now,
%% \begin{eqnarray}
%%  \lim_{R \to \infty} \int_{\mathbb R} |u_x|^2 e^{cx} \eta_x dx &=& 0,
%%  \nonumber \\ \lim_{R \to \infty} \int_{\mathbb R} \{ (1 - \mu)
%%  \frac{|u_x|^2}{2} + V(u) \} \eta_x e^{cx} &=& 0, \nonumber \\ \lim_{R
%%  \to \infty} \int_{-R}^{R} \left\{ (1 - \mu) \frac{|u_x|^2}{2} +
%%  V(u) \right\} e^{cx} \eta &=& \int_{\mathbb R} \left\{ (1 - \mu)
%%  \frac{|u_x|^2}{2} + V(u) \right\} e^{cx}.
%% \end{eqnarray}
%% Since $u \in {\mathcal B}$, we deduce $\Phi_c (u) - \mu = 0$. 

\end{proof}


We are now ready to prove Proposition \ref{travex}. By Lemmas
\ref{l01} and \ref{weak}, the solution $u = u_c$ of problem (P)
satisfies
\begin{eqnarray*}
\label{el}
(1 - \mu) (u_{xx} + c u_x) + f(u) = 0,
\end{eqnarray*}
where $\mu \leq 0$ is given by Eq.~(\ref{lam}). Introducing $\bar
u(x) = u_c(x \sqrt{1 - \mu})$ and using Eq.~(\ref{el}), we obtain
that $\bar u$ is a traveling wave with speed $c^* = c \sqrt{1 -
\mu}$. Furthermore, $\bar u \in [0, 1]$ and $\bar u \in
H^1_{c^*}(\mathbb R)$. Indeed, with $u = u_c$ we have
\begin{eqnarray*}
\int_\mathbb{R} e^{c^*x} \bar u^2 dx = \int_\mathbb{R} e^{c x \sqrt{1
- \mu}} u^2( x \sqrt{1 - \mu}) dx = {1 \over \sqrt{1 - \mu}}
\int_\mathbb{R} e^{c x} u^2 dx, \\ \int_\mathbb{R} e^{c^*x} \bar u_x^2
dx = \int_\mathbb{R} e^{c x \sqrt{1 - \mu}} u_x^2( x \sqrt{1 - \mu})
dx = \sqrt{1 - \mu} \int_\mathbb{R} e^{c x} u_x^2 dx.
\end{eqnarray*}
Now, by Proposition \ref{p:volpert} any traveling wave solution
bounded between 0 and 1 is one of the solutions in Proposition
\ref{awtrav}. Furthermore, by Proposition \ref{awtrav}, for $c > c^*$
any traveling wave solution with the asymptotic decay given by
Eq.~(\ref{lam-}) is not in $H^1_c(\mathbb R)$, so we are left with the
only possibility that $\bar u$ is the traveling wave solution
corresponding to Eq.~(\ref{lam+}). \qed

\begin{remark}
We point out that the speed $c^*$ of the obtained traveling wave is
independent of the value of $c$ in Proposition \ref{travex}. This is
in fact a general property of the considered variational problem (for
more details, see \cite{m,hlmn}).
\end{remark}

\begin{remark}
Observe that by Proposition \ref{travex} we have $c^* \geq c$, so the
value of $c$ in Theorem \ref{main} also provides the lower bound for
the propagation speed.
\end{remark}


To show that the assumption of Theorem \ref{main} is necessary, we
note simply that if $c^* > c_0$ in Proposition \ref{awtrav}, then by
Eq.~(\ref{lam+}) and the fact that $|\bar u_x| \leq C \bar u$ with
some $C$ for the traveling wave solution $\bar u$ with speed $c^*$
(see \cite{aronson75}), we have $\bar u \in H^1_{c^*}(\mathbb
R)$. Furthermore, $\bar u$ satisfies Eq.~(\ref{el}) with $c = c^*$ and
$\mu = 0$. This implies that $\Phi_{c^*}[\bar u] = 0$, which gives the
desired result.

\section{Existence of minimizers} \label{s:exist}

We now show that under the assumption of Theorem \ref{main} problem
(P) always has a solution. To prove that, we need to establish a few
auxiliary results regarding the properties of the functional
$\Phi_c$. First, the following Lemma is of fundamental importance to
the study of problem (P) and gives the analog of the Poincar\'e
inequality for spaces $H^1_c(\mathbb R)$.

\begin{lemma}
\label{l:Hardy}
For all $u \in H^1_c (\mathbb R)$, we have
%
\begin{equation} \label{eq:Hardy}
 \frac{c^2}{4} \int_{R}^{\infty} e^{cx} u^2 dx \leq \int_R^{\infty}
 e^{cx} u_x^2 dx \qquad \forall R \in \mathbb R,
\end{equation}
%
\begin{equation}  \label{eq:Hardy2}
 u^2(R) \leq \frac{e^{-c R}}{c} \int_R^{\infty} e^{cx} u_x^2 dx \qquad
 \forall R \in \mathbb R.
\end{equation}
\end{lemma}

\begin{proof}
Let us first prove Eq.~(\ref{eq:Hardy}). We have
\begin{eqnarray*} 
\frac{c}{2} \int_R^{\infty} e^{cx} u^2 dx &=& -\frac{1}{2} e^{c R} u^2
(R) - \int_R^{\infty} e^{cx} u u_x dx \\ & \leq & \left(
\int_{R}^{\infty} e^{cx} u^2 dx \right)^{1/2} \left( \int_R^{\infty}
e^{cx} u_x^2 dx\right)^{1/2},
\end{eqnarray*}
which implies (\ref{eq:Hardy}).

Let us now prove Eq.~(\ref{eq:Hardy2}). Since, $ \int_R^{\infty}
e^{cx} \left( \sqrt c u + \frac{1}{\sqrt c} u_x \right)^2 dx \geq 0$,
we get
\begin{eqnarray*}
 \frac{1}{c} \int_R^{\infty} e^{cx} u_x^2 dx + c \int_R^{\infty}
 e^{cx} u^2 dx &\geq& -2 \int_{R}^{\infty} e^{cx} u u_x dx \\ &=&
 u^2(R) e^{cR} + c \int_R^{\infty} e^{cx} u^2 dx,
\end{eqnarray*}
which gives (\ref{eq:Hardy2}).   
\end{proof}

An immediate consequence of Eq.~(\ref{eq:Hardy}) is 
\begin{corollary}
If $u \in H^1_c(\mathbb R)$, then
\begin{eqnarray}
\label{poinc}
{c^2 \over 4} \int_\mathbb{R} e^{cx} u^2 dx \leq \int_\mathbb{R}
e^{cx} u_x^2 dx.
\end{eqnarray}
\end{corollary}

Observe that since under our assumptions $V(u) \geq - C u^2$ with some
$C \geq 0$, the functional $\Phi_c$ will be positive for all non-zero
$u \in H^1_c(\mathbb R)$ for sufficiently large $c$. More precisely,
we have (see also \cite{m})
\begin{lemma}
\label{l:cmax}
Let $c_\mathrm{max}$ be defined as
\begin{eqnarray}
\label{cmax}
c_\mathrm{max} = \min \left\{ c \geq 0: \quad \frac{1}{8} c^2 s^2 +
V(s) \geq 0, \quad \forall s \in \mathbb R \right\}.
\end{eqnarray}
Then, for any $c > c_\mathrm{max}$ we have $\Phi_c[u] > 0$ for all $u
\in H^1_c(\mathbb R)$, such that $u \not \equiv 0$.
\end{lemma}
Combined with Theorem \ref{main}, this result gives an upper bound for
the value of $c^*$. 

\begin{proposition}
\label{p:cmax}
The value of $c_\mathrm{max}$ in Eq.~(\ref{cmax}) gives an upper bound
for the propagation speed $c^*$ in Theorem \ref{awspeed}.
\end{proposition}

\begin{proof}
First of all, we note that $c_\mathrm{max} < \infty$. Secondly, we
must have $c_\mathrm{max} \geq c_0$. Indeed, we have
$$
V(s) \sim - \frac{1}{2} f'(0) s^2, \qquad s \to 0^+,
$$
with $f'(0) > 0$.  Recalling the definition of $c_0$ in
Eq.~(\ref{c0}), we see that $c_\mathrm{max} \geq c_0$ in order for the
inequality in Eq.~(\ref{cmax}) to hold for small $s$. Now, if $c^* >
c_\mathrm{max}$, then $c^* > c_0$, so by the argument at the end of
section \ref{s:constr} we have $\Phi_{c^*}[\bar u] = 0$, where $\bar
u$ is the traveling wave solution with speed $c^*$, contradicting
Lemma \ref{l:cmax}.
\end{proof}

\begin{remark}
The statement of Proposition \ref{p:cmax} remains valid in a much more
general context (see \cite{m}).
\end{remark}

Before proceeding further to the proof of existence of solutions of
problem (P), let us introduce the following notation. For given
$-\infty \leq a < b \leq +\infty$, define
\begin{eqnarray}
\Phi_c[u, (a, b)] = \int_a^b e^{cx} \left( \frac{1}{2} u_x^2 + V(u)
\right) dx.
\end{eqnarray}

\begin{lemma}
\label{l:c}
Assume $c > c_0$, then there exists $R > 0$ such that 
\begin{enumerate}
\item 
$\Phi_c [u, (R, +\infty)] \geq 0$ for all $u \in \mathcal B$;
\item
letting $u_n \rightharpoonup u$ in $H^1_c(\mathbb R)$, then
$\liminf_{n \to \infty} \Phi_c [u_n, (-\infty,R)] \geq \Phi_c [u,
(-\infty, R)].  $
\end{enumerate}
\end{lemma}
\begin{proof} 
1) Since $c > c_0$, we can choose $0 < \epsilon < \frac{c^2}{4} -
f'(0)$.  By the definition of $V$, there exists some $s_0 > 0$ such
that
$$
  V(s) \geq -\frac{1}{2}(f'(0) + \epsilon) s^2 , \qquad |s| \leq s_0.
$$
 From Lemma \ref{l:Hardy}, there exists $R_0$ such that
$$
  |u(x)| < s_0 \qquad \forall x > R_0, \quad \forall u \in \mathcal B.
$$
Therefore, given $R > R_0$, we get
\begin{equation}
  \int_R^{\infty} e^{cx} V(u) dx \geq - \frac{1}{2} (f'(0) + \epsilon)
  \int_{R}^{\infty} e^{cx} u^2dx.  \label{eq:Ineq}
\end{equation}
%
Thus, from Eqs.~(\ref{eq:Hardy}) and (\ref{eq:Ineq}) and the choice of
$\epsilon$, we obtain
$$
  \Phi_c [u, (R, +\infty)] \geq \frac{1}{2} \left( \frac{c^2}{4} -
  f'(0) - \epsilon \right) \int_R^{\infty} e^{cx} u^2 dx \geq 0,
$$
which concludes the proof.

2) Since $V(u)$ is bounded from below, this follows by standard
semicontinuity results, see, for example, \cite[Propositions 2.1,
2.2]{DM}.
\end{proof}

We are now ready to prove our existence result. We note that our
method has a number of features in common with the technique developed
by Berestycki and Lions for scalar field equations
\cite{berestycki83}. Similar techniques were also used by Heinze in
\cite{heinze}.

\begin{proposition}
\label{exist}
Suppose there exists $u \in \mathcal B$ such that $\Phi_c[u] \leq 0$
for some $c > c_0$. Then problem (P) has a solution.
\end{proposition}

\begin{proof}
Let $(u_n)$ be a minimizing sequence of Problem (P), i.e $u_n \in
\mathcal B$ with $\Phi_c [u_n] \to \inf_\mathcal{B} \Phi_c$.  By
assumption, $\inf_\mathcal{B} \Phi_c \leq 0$. Since $\Gamma_c[u_n] =
1$, from inequality (\ref{poinc}), we get that $\int_\mathbb{R} e^{cx}
u_n^2 dx \leq \frac{8}{c^2}$. Thus, $(u_n)$ is bounded in $H_c^1
(\mathbb R)$ and therefore it converges weakly to some $u \in H_c^1
(\mathbb R)$. Furthermore,
\begin{eqnarray}
  \inf_\mathcal{B} \Phi_c & = & \liminf_{n \to \infty} \Phi_c [u_n]
  \nonumber \\ & \geq & \liminf_{n \to \infty} \{ \Phi_c [u_n,
  (-\infty, R)] \} + \liminf_{n \to \infty} \{ \Phi_c [u_n,
  (R,+\infty)] \} \nonumber \\ & \geq & \Phi_c [u, (-\infty, R)]
  \nonumber \\ & = & \Phi_c [u] - \Phi_c [u, (R, +\infty)],
  \label{eq:Decrease}
\end{eqnarray}
for large enough $R$. Now, by letting $R \to + \infty$ and noting that
$\displaystyle\lim_{R \to +\infty} \Phi_c [u,(R, +\infty)] = 0$,
Eq.~(\ref{eq:Decrease}) leads to
\begin{equation*}
0 \geq \inf_\mathcal{B} \Phi_c \geq \Phi_c [u].
\end{equation*}
If $\, \inf_\mathcal{B} \Phi_c = 0$, we deduce that $u$ in the
assumption of this proposition is a minimizer. Therefore, let us
assume that $\inf_\mathcal{B} \Phi_c < 0$. Then $u \not \equiv 0$, and
since
$$
  1 = \liminf_{n \to \infty} \Gamma_c [u_n] \geq \Gamma_c [u] > 0,
$$
we can, by using Lemma \ref{lem:Translation}, 
choose $a \geq 0$ such that
$$
  \Gamma_c [u_a] = 1 \quad \hbox{ with } \quad u_a (x) = u(x - a).
$$
Since $\inf_\mathcal{B} \Phi_c \leq 0$ and $a \geq 0$, we derive
$$
  \Phi_c [u_a] = e^{ca} \Phi_c [u] 
  \leq \Phi_c [u] 
  \leq \inf_\mathcal{B}\Phi_c.
$$
Therefore, $\Phi_c [u_a] = \inf_\mathcal{B} \Phi_c$, and $u_a$ solves
Problem (P). 

\end{proof}

Theorem \ref{main} then follows by noting that if $\Phi_c[u] \leq 0$
for some $u \not \equiv 0$, then, according to Lemma
\ref{lem:Translation}, we can make $u \in \mathcal B$ by an
appropriate shift.

\begin{remark}
It is not difficult to see that Propositions \ref{travex} and
\ref{exist} remain valid when $f(u)$ is not necessarily positive for
all $u \in (0, 1)$.
\end{remark}

Thus, our method also provides a general method of constructing the
traveling wave solutions, as well as obtaining upper and lower bounds
for their speed. Moreover, in the case of $f'(0) \leq 0$ there is no
need for any assumptions on $c$ (since $c^2 - 4 f'(0) > 0$ in Lemma
\ref{l:c} for all $c > 0$), and the assumption of Proposition
\ref{exist} is always satisfied as long as $V(u) < 0$ for some $0 < u
\leq 1$ (see section \ref{s:appl}).

\section{Some applications} \label{s:appl}

In this section, we first give two results concerning sufficient
conditions for linear and nonlinear selection mechanisms,
respectively. We then perform a variational study of a particular
example, namely Eq.~(\ref{quint}).

Let us first formulate the general sufficient condition for the linear
selection mechanism. 

\begin{theorem}
If for all $0 < u \leq 1$
\begin{eqnarray}
\label{linsuf}
{2 \over u^2} \int_0^u f(s) ds \leq f'(0),
\end{eqnarray}
then the linear selection is realized.
\end{theorem}

\begin{proof}
 From Eq.~(\ref{linsuf}) we obtain
$$
V(u) \geq -\frac{1}{2} f'(0) u^2 = -{1\over 8} c_0^2 u^2,
$$
so $c_\mathrm{max} = c_0$ in Eq.~(\ref{cmax}). Therefore, by Lemma
\ref{l:cmax} we have $\Phi_c[u] > 0$ for all $u \not \equiv 0$ and $c
> c_0$. The result then follows from the ``only if'' statement of
Theorem \ref{main}.
\end{proof}

Naturally, this implies that $c^* = c_0$. Observe that this result is
the generalization of that of Kolmogorov, Petrovsky, and Piskunov, who
required that $f'(u) \leq f'(0)$ \cite{kolmogorov37}, as well as the
result of Aronson and Weinberger, who obtained an upper bound for
$c^*$, which is equal to $c_0$ if $f(u)/u \leq f'(0)$
\cite{aronson78}.

We now give a sufficient criterion of the nonlinear selection
mechanism. We note that better criteria can be obtained for a given
nonlinearity by using suitable trial functions (see below). This
criterion, nevertheless, gives a more precise meaning to the ``ZFK''
case considered in \cite{roquejoffre97} and is relevant to combustion.

\begin{theorem}
\label{t:nonlinsuf}
If
\begin{eqnarray}
\label{nonlinsuf}
f'(0) \leq \frac{1}{2} \int_0^1 f(u) du,
\end{eqnarray}
then the nonlinear selection mechanism is realized.
\end{theorem}

\begin{proof} 
Let us take
\begin{eqnarray*}
u_\lambda (x) =
\begin{cases} 
1, & x \leq 0, \\ 
e^{-\lambda x}, & x > 0.
\end{cases}
\end{eqnarray*}
Then for $u = u_\lambda(x)$ we have
\begin{eqnarray*}
\Phi_c[u] < \frac{1}{2} \int_0^\infty e^{cx} u_x^2 dx +
\int_{-\infty}^0 e^{cx} V(u) dx \nonumber \\ = {\lambda^2 \over 2 (2
\lambda - c)} + {V(1) \over c},
\end{eqnarray*}
as long as $\lambda > {c \over 2}$. Minimizing this expression with
respect to $\lambda$, we obtain that the minimum is achieved at
$\lambda = \lambda_\mathrm{min} = c$. This means that
\begin{eqnarray*}
\Phi_c[u_{\lambda_\mathrm{min}}] < {c \over 2} + {V(1) \over c}.
\end{eqnarray*}
Recalling that $V(1) < 0$, we see that the expression above is
negative whenever 
\begin{eqnarray}
\label{cnarrow}
c = c_\mathrm{min} = \sqrt{-2 V(1)} = \sqrt{2 \int_0^1 f(u) du}.
\end{eqnarray}
By continuity, there exists $c > c_\mathrm{min}$ such that
$\Phi_c[u_{\lambda_\mathrm{min}}] \leq 0$, so $c > c_0$ (see
Eq.~(\ref{c0})) and $u_{\lambda_\mathrm{min}}$ satisfies the
assumption of Theorem \ref{main}.
\end{proof}

Note that the expression in the right-hand side of Eq.~(\ref{cnarrow})
tends to $c^*$ in the limit of narrow reaction zone (that is, when
$f(u)$ is concentrated around $u = 1$). Of course, it also provides a
lower bound for $c^*$. Also note that for general $f(u)$ (not
necessarily positive) this proof can be modified to obtain an
analogous estimate, as long as $V(1) < 0$, which is a necessary
condition for the existence of traveling waves with $c > 0$ (see, for
example, \cite{volpert}).

Let us now perform a variational study of Eq.~(\ref{quint}). This
equation with $\mu > 0$ satisfies the assumptions in Eq.~(\ref{ass})
after a trivial rescaling of $u$. The stable positive equilibrium $u =
u_\mathrm{max}$ is given by
\begin{eqnarray}
\label{umax}
u_\mathrm{max} = {\sqrt{1 + \sqrt{1 + 4 \mu}} \over \sqrt{2}}.
\end{eqnarray}
The nice thing about Eq.~(\ref{quint}) is that it admits a traveling
wave solution with speed $c > 0$ for $\mu > -\frac{3}{16}$ which can
be found exactly \cite{vansaarloos89,powell91}. Furthermore, for $0 <
\mu < \frac{3}{4}$ this is precisely the solution with speed $c^*$,
which is explicitly given by
\begin{eqnarray}
\label{csq}
c^* = {2 \sqrt{1 + 4 \mu} - 1 \over \sqrt{3}}.
\end{eqnarray}
Numerical evidence also suggests that for $\mu \geq \frac{3}{4}$ we
have $c^* = c_0 = 2 \sqrt{\mu}$. Note that the expression in
Eq.~(\ref{csq}) for the speed of the wave also remains valid for $\mu
< 0$, it corresponds to the unique (up to translations) traveling wave
solution in this case \cite{aronson75}. Also, when $\mu = 0$, it
corresponds to the unique traveling wave solution which decays
exponentially at plus infinity (for an existence proof in this
degenerate case, see \cite{medvedev,billingham00}). Since our proof of
existence of the traveling wave solution in fact extends to the case
of $f'(0) \leq 0$, we will treat Eq.~(\ref{quint}) for all $\mu >
-\frac{3}{16}$. In short, the availability of an analytical expression
for the speed $c^*$ makes it possible to assess the effectiveness of
our variational method, as well as study the parametric dependence of
$c^*$.

We start by looking for the value of $c_\mathrm{max}$ as a function of
$\mu$. Observe that for $0 \leq u \leq 1$ we have
\begin{eqnarray*}
\frac{1}{8} c^2 u^2 + V (u) = \frac{1}{2} \left( {c^2 \over 4} - \mu -
\frac{1}{2} u^2 + \frac{1}{3} u^4 \right) u^2.
\end{eqnarray*}
So, minimizing the expression in the brackets and demanding that it
remains positive for all $u$, we obtain
\begin{eqnarray}
\label{cmaxq}
c_\mathrm{max} = \frac{1}{2} \sqrt{ 3 + 16 \mu}.
\end{eqnarray}
Note that the value of $c_\mathrm{max}$ in Eq.~(\ref{cmaxq}) provides
an upper bound for $c^*$ and is typically within $\sim 20$\% of the
exact value in Eq.~(\ref{csq}). This equation also shows that
$c_\mathrm{max} > 0$ only for $\mu > -\frac{3}{16}$, as expected from
the exact solution.

We are now going to analyze the existence of the traveling wave
solutions with speed $c^* > c_0$ and obtain the lower bounds for the
speed, using our variational approach. To proceed, let us choose a
very simple trial function that looks like a front:
\begin{eqnarray}
\label{ul}
u_\lambda(x) = 
\begin{cases}
\frac{1}{2} u_\mathrm{max} e^{-\lambda x}, & x \geq 0, \\ \frac{1}{2}
u_\mathrm{max} \left( 2 - e^{\lambda x} \right), & x < 0,
\end{cases}
\end{eqnarray}
where $u_\mathrm{max}$ is given by Eq.~(\ref{umax}). So, $u_\lambda$
is a $C^1$ function connecting the unstable equilibrium $u = 0$ at
plus infinity with the stable equilibrium $u = u_\mathrm{max}$ at
minus infinity, characterized by just a single parameter $\lambda$
that gives the rate of exponential decay at plus infinity. What we are
going to show below is that this choice of the trial function already
allows to determine the value of $c^*$ with an accuracy of just a few
per cent in the entire range of $\mu$.

Let us substitute $u_\lambda$ into the functional. It is
straightforward, although tedious, to perform the integration, the
resulting expression is a rational function of $\lambda > {c \over 2}$
(the algebraic calculations were performed using \textsc{Mathematica}
software):
\begin{eqnarray*}
&& \Phi_c[u_\lambda] = \nonumber \\ && -( {\lambda }^2( 16c^7\lambda (
         1 + {\sqrt{1 + 4\mu }} ) - 92160{\lambda }^6( 1 + {\sqrt{1 +
         4\mu }} + \mu ( 6 + 4{\sqrt{1 + 4\mu }} ) ) \nonumber \\ && -
         8c^4{\lambda }^2( 9( 1 + {\sqrt{1 + 4\mu }} ) + 832{\lambda
         }^2( 1 + {\sqrt{1 + 4\mu }} ) - 3\mu ( 13 + 19{\sqrt{1 + 4\mu
         }} ) ) \nonumber \\ && + 192c{\lambda }^5( 53( 1 + {\sqrt{1 +
         4\mu }} ) + 720{\lambda }^2( 1 + {\sqrt{1 + 4\mu }} ) - 2\mu
         ( 222 + 275{\sqrt{1 + 4\mu }} ) ) \nonumber \\ && +
         48c^2{\lambda }^4( 295( 1 + {\sqrt{1 + 4\mu }} ) +
         1536{\lambda }^2( 1 + {\sqrt{1 + 4\mu }} ) + \mu ( 973 +
         383{\sqrt{1 + 4\mu }} ) ) \nonumber \\ && - 12c^3{\lambda
         }^3( -187( 1 + {\sqrt{1 + 4\mu }} ) + 272{\lambda }^2( 1 +
         {\sqrt{1 + 4\mu }} ) - \mu ( 1153 + 779{\sqrt{1 + 4\mu }} ) )
         \nonumber \\ && - c^5\lambda ( 592{\lambda }^2( 1 + {\sqrt{1
         + 4\mu }} ) + 9( 5 + 27\mu + 5{\sqrt{1 + 4\mu }} + 17\mu
         {\sqrt{1 + 4\mu }} ) ) \nonumber \\ && - c^6( -128{\lambda
         }^2( 1 + {\sqrt{1 + 4\mu }} ) + 3( 1 + {\sqrt{1 + 4\mu }} +
         \mu ( 7 + 5{\sqrt{1 + 4\mu }} ) ) ) ) )/ \nonumber \\ && (
         64c( c - 6\lambda ) ( c - 4\lambda ) ( c - 2\lambda ) ( c +
         2\lambda ) ( c + 3\lambda ) ( c + 4\lambda ) ( c + 5\lambda )
         ( c + 6\lambda ) ).
\end{eqnarray*}
The situation is complicated somewhat by the $\mu$-dependence, so
before we apply our techniques we need to make sure that
$\Phi_c[u_\lambda]$ can in fact be negative in a certain range of
$\mu$ for $c > c_0 = 2 \sqrt{\mu}$. To do this, we will use the
following trick. Take a sufficiently small $\epsilon > 0$ and consider
$\lambda_\epsilon = \sqrt{\mu} + \epsilon$ and $c = 2 \sqrt{\mu} +
\epsilon^2$. Then clearly $u_{\lambda_\epsilon} \in
H^1_{c_\epsilon}(\mathbb R)$. A direct calculation then shows that
\begin{eqnarray}
\label{lep}
\lim_{\epsilon \to 0^+} \Phi_{c_\epsilon} [u_{\lambda_\epsilon}] =
    {\mu \left( 331 + 5785{\sqrt{1 + 4\mu }} \right) -2727 \left( 1 +
    {\sqrt{1 + 4 \mu }} \right) \over 215040{\sqrt{\mu }}}.
\end{eqnarray}
Analyzing this expression as a function of $\mu$, we see that it is
negative as long as $0 < \mu < \mu_\mathrm{max}$, where
\begin{eqnarray}
\label{mmax}
\mu_\mathrm{max} = {23212224 \over 33466225} \simeq 0.6936. 
\end{eqnarray}
Now, since the limit in Eq.~(\ref{lep}) exists, for all $0 < \mu <
\mu_\mathrm{max}$ it is possible to choose $\epsilon$ such that
$\Phi_{c_\epsilon} [u_{\lambda_\epsilon}] \leq 0$, so
$u_{\lambda_\epsilon}$ satisfies the assumption of Theorem
\ref{main}. Thus, we have just proved, using the trial function in
Eq.~(\ref{ul}), that the nonlinear selection mechanism is realized at
least for $0 < \mu < \mu_\mathrm{max}$, with $\mu_\mathrm{max}$ given
explicitly by Eq.~(\ref{mmax}). Note that numerical analysis of the
expression for $\Phi_{c_\epsilon} [u_{\lambda_\epsilon}]$ shows that
the value of $\mu_\mathrm{max}$ obtained above is in fact
optimal. Comparing this with the known exact result, we see that
$\mu_\mathrm{max}$ slightly underestimates the maximal value of $\mu =
\frac{3}{4}$ below which the nonlinear selection mechanism is
realized \cite{vansaarloos89,powell91}.

Let us now look at how closely our variational procedure allows to
estimate the propagation speed $c^*$ from below. First of all, note
that $c_0 = 2 \sqrt{\mu}$ is a natural lower bound for the speed $c^*$
for $\mu > 0$. In fact, for $\mu \geq \mu_\mathrm{max}$ the difference
between $c^*$ and $c_0$ is less than 0.03\%. Now, for $\mu <
\mu_\mathrm{max}$ and fixed $c$, $\Phi_c[u_\lambda]$ as a function of
$\lambda$ has a minimum for $\lambda > {c \over 2}$, since
$\Phi_c[u_\lambda] \to +\infty$ as $\lambda \to {c \over 2} + 0$
(since in this case the integral is dominated by the quadratic terms
at plus infinity, which for $c > c_0$ are positive) or $\lambda \to
+\infty$. Therefore, we look for the values of $c$ and $\lambda$ that
solve
\begin{eqnarray*}
\Phi_c[u_\lambda] = 0, \qquad {\partial \Phi_c[u_\lambda] \over
\partial \lambda} = 0.
\end{eqnarray*}
The solution is obtained numerically for all $-\frac{3}{16} < \mu <
\mu_\mathrm{max}$. The results are shown in Fig. \ref{f:c}.
\begin{figure}
\centerline{\psfig{figure=fig1.eps,width=8cm}}
\caption{Speed of the traveling wave as a function of $\mu$.}
\label{f:c}
\end{figure}
This figure shows the speed $c$ obtained above, together with the
exact speed $c^*$ given by Eq.~(\ref{csq}), as well as
$c_\mathrm{max}$ given by Eq.~(\ref{cmaxq}) and $c_0$, all as
functions of $\mu$. Note that the dependences $c(\mu)$ and $c^*(\mu)$
are virtually indistinguishable. To see the agreement between $c$ and
$c^*$, we plot the difference between the two as a function of $\mu$
in Fig. \ref{f:d}.
\begin{figure}
\centerline{\psfig{figure=fig2.eps,width=8cm}}
\caption{The absolute error $c^* - c$ as a function of $\mu$.}
\label{f:d}
\end{figure}
For $\mu \geq \mu_\mathrm{max}$, we use $c = c_0$ as the lower bound
for $c^*$. One can see that the error is within 1\% of the true value
for all $\mu > 0$ and rapidly decreases as $\mu$ increases. We also
verified that the speed $c$ remains within 2.5\% of the exact value in
the {\em entire} range of $\mu$ (including $-\frac{3}{16} < \mu \leq
0$). This accuracy is quite remarkable, especially considering the
fact that it is achieved by tuning only one parameter, $\lambda$, in
the trial function. Note that we experimented with a number of
different choices of the trial functions, they all give comparable
results.  Thus, our method provides a very efficient way of estimating
the propagation speed, as well as establishing the ranges of
parameters for the nonlinear selection mechanism.



\section{Concluding remarks} \label{s:concl}

To conclude, we have obtained a variational characterization of
certain traveling wave solutions for scalar reaction-diffusion
equations. These traveling waves are special in the sense that they
have a non-generic exponential decay at plus infinity (see
Eq.~(\ref{lam+})). In the context of the theory of Aronson and
Weinberger these solutions determine the propagation speed for the
sufficiently localized initial conditions, so our method gives an
easily verifiable answer to the problem of linear vs. nonlinear
selection mechanism. We showed that our method allows to obtain very
accurate lower bounds for the propagation speed, as well as simple
upper bounds.

We remark that various other variational characterizations of the
speed $c^*$ exist. One method is based on the minimax characterization
of the traveling wave solutions (see, for example,
\cite{volpert}). Another technique, introduced by Benguria and
Depassier, uses an integral variational principle in the phase plane
\cite{benguria96}. We point out, that these methods are formulated
within phase plane and may therefore be rather sensitive to the
choices of the trial functions. We, on the other hand, demonstrated
that in our variational approach there is very little sensitivity to
the choice of the trial functions, what allows very accurate estimates
of the propagation speed, at least for the example considered
above. We also point out that our method does not just characterize
the speed $c^*$, but also gives an alternative proof of existence of
the traveling wave solutions of certain types, and is in fact much
more general than the original setup (see also \cite{hlmn}).

Concerning the problem of propagation, the framework of Aronson and
Weinberger allows to associate the speed $c^*$ with the average
asymptotic speed of the leading edge for the solutions of
Eq.~(\ref{rd}) with suitable initial conditions. An interesting
question arising here is whether this remains true for more general
classes of initial conditions, for example, in the case of
Eq.~(\ref{quint}), for initial conditions that are not necessarily
positive (see, for example, \cite{powell91}). Such problems have
recently been addressed in \cite{m}, where the generalization of the
functional from Eq.~(\ref{phi}) was used to study questions of
propagation and convergence to traveling wave solutions for
reaction-diffusion systems of gradient type in cylinders in a very
general setting. It is interesting to see whether the selection
problem discussed in this paper is meaningful in this more general
context and in fact reduces to the problem of existence of the
special traveling waves whose existence can be captured by our
variational procedure.


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\end{document}
