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\begin{document}
\begin{center}
\begin{large}
{\bf Variational Characterization of the Speed of Propagation of
\\ Fronts
for the Nonlinear Diffusion Equation}
\end{large}
\end{center}
\vspace{.1cm}
\begin{center}
R.\ D.\ Benguria and M.\ C.\ Depassier\\
Facultad de F\'\i sica\\
P. Universidad Cat\'olica de Chile\\
Casilla 306, Santiago 22, Chile
\end{center}
\date\today
\begin{abstract}
We give a variational characterization for the speed of fronts
of the nonlinear diffusion equation $u_t = u_{xx} + f(u)$ with
$f>0$, which permits the calculation of the exact speed for
arbitrary $f$.
\end{abstract}
\newpage
\section{Introduction}
The problem of the asymptotic speed of propagation of the
interface between an unstable and stable state has received much
attention in connection with different problems of population
growth, chemical reactions, pattern formation and others. We refer to
\cite{CH93} for a recent review and references. The
best understood of such problems is that of the nonlinear
reaction diffusion equation
$$ u_t = u_{xx} + f(u) \eqno(1a) $$
with
$$ f(0) = f(1) = 0,\,\, f'(0) > 0 \qquad {\rm and}\quad
f>0 \qquad {\rm in}\quad (0,1) \eqno(1b)
$$
for which Aronson
and Weinberger [AW]\cite{AW78} have shown that any positive
sufficiently localized initial condition $u(x,0)$ evolves into a
front that joins the stable state $u=1$ to $u=0$. The asymptotic
speed at which the front propagates is the minimal speed $c^*$
for which there is a monotonic front joining $u=1$ to $u=0$.
Moreover they show that the selected speed is bounded above and
below by
$$ 2\sqrt{f'(0)} \le c^* < 2 \sup\sqrt{f(u)\over u} \eqno(2)
$$
and that the asymptotic selected front approaches
the fixed point $u=0$ exponentially with slope
$$
m= -{1\over2} (c^* + \sqrt{c^{*2}-4f'(0)}). \eqno(3)
$$
The lower bound
$c_L = 2\sqrt{f'(0)}$ is that predicted from a physical
argument, the linear marginal stability hypothesis\cite{KPP37}.
For concave functions $f$, the upper and lower bounds coincide
and the speed is exactly the linear value. However, the
asymptotic speed of propagation can still be the linear value
even when the upper and lower bounds do not coincide as explicit
examples and a variational characterization\cite{HR75} which
provides improved upper bounds shows. We have recently obtained
an improved lower bound\cite{BD94} on the speed of the front
that enables one to decide when the selected speed is greater than the linear
value case which is referred to as nonlinear marginal stability selection.
There have been several reformulations of Aronson's and
Weinberger's rigorous results for the nonlinear diffusion
equation aiming to their heuristic extension to other higher order and
pattern forming equations\cite{DL83,BBDKL85,VS89,PCGO94}.
None of these approaches however
provide the means to calculate a priori the velocity of the fronts.
The purpose of the present work is to extend our previous result\cite{BD94}
to show a variational characterization of the speed of the
fronts of equation (1) which enables its exact calculation for
arbitrary $f$. Our main result is
$$
c^* = \left\{ \begin{array}{ll}
\max\left\{I(g)|g\in \cal D\right\}
& \mbox{if $c^* > 2\sqrt{f'(0)}$} \\
2\sqrt{f'(0)} &\mbox{otherwise}
\end{array} \right. \eqno(4)
$$
where
$$
I(g) = 2 {\int_0^1 \sqrt{f g h} du \over \int_0^1 g du} \eqno(5)
$$
and
$\cal D$ is the space of functions such that
$$
g \ge 0, \quad h = - g' > 0 \quad g(1) = 0 \qquad {\rm and}
\int_0^1 g(u)\, du < \infty \eqno(6)
$$
In Section 2 we prove this result and in Section 3 an example is given.
\section{The Variational Characterization}
We are interested in the calculation of the minimal speed for which
equation (1) has a monotonic travelling front $u(x,t) = q (z)$ with
$z =x - ct$
joining $u=1$ to $u =0$.
Since the selected speed corresponds to that of a decreasing monotonic
front, it is convenient to work in phase space.
Calling
$p(q) = - dq/dz$, where the minus sign is included so that $p$ is positive, we
find that the monotonic fronts are solutions of
$$
p(q)\, {dp\over dq} - c^*\, p(q) + f(q) = 0, \eqno(7)
$$
with
$$
p(0) = 0, \qquad p(1) = 0, \qquad p > 0 \quad {\rm in}\quad (0,1). \eqno(8)
$$
In the derivation of the main result we shall make use of the known\cite{AW78}
asymptotic behavior of $p$ at the origin which we briefly recall.
Near $q=0$, since the
approach to the fixed point is exponential and given by (3 ),
$p \sim a_1 q$ with $a_1 = - m$.
This value of $a_1$ is the largest root
of
$$
a_1^2 - c a_1 + f'(0) = 0. \eqno(9)
$$
We find it convenient to introduce the parameter $\lambda$ defined by
$$
c = \lambda \, a_1
$$
then,
$$
c = \lambda \sqrt{{f'(0)\over \lambda -1 }}\qquad {\rm and}\qquad a_1 =
\sqrt{{f'(0)\over \lambda -1}} \eqno(10)
$$
It is straightforward to verify that whenever $1< \lambda < 2$ the
above value of $a_1$ corresponds to the largest root of (9) and therefore
to the asymptotic slope at the origin of the selected front\cite{BD94A}. At
$\lambda =2$ the speed attains the linear value $c_L$.
For completeness we shall repeat part of the developments given in\cite{BD94}.
Let $g$ be any function in $\cal D$ defined in (6).
Multiplying equation (7) by $g/p$ and integrating with respect to $q$ we find
after integrating by parts,
$$
c^* = {\int_0^1 \left( h\, p + {f(q)\over p}\, g \right) dq\over
\int_0^1 g(q) dq}
\eqno(11)
$$
However since $p,\,h,\, f, {\rm and}\, g$ are positive,
for every fixed $q$
$$
h\,p + {f(q)\, g\over p} \ge 2 \, \sqrt{f\, g\, h} \eqno(12)
$$
hence
$$
c^*\, \ge 2\, {{\int_0^1 \sqrt{ f\, g\, h}\, dq}\over{\int_0^1
g\, dq}} \eqno(13)
$$
In order to prove the main result (4), we must show that we can
always find a function $g$ for which the equality in (12) and
therefore in (13), holds.
Let $v(q)$ be the positive solution of the equation
$$
{ v'\over v} = {c^*\over p} \eqno(14)
$$
and choose
$$
g = {1\over v'}. \eqno(15)
$$
First we show that this function belongs to $\cal D$. Since $p$ is positive
there is always a positive solution for $v$ and from (14) we see that
$ v'$ and therefore $g$ is positive. Also one can always adjust
the constant of integration in (14) in such a way to have $g(1) = 0$.
We must verify that $h = -g' > 0$. From its definition
$$
g' = - { v'' \over v'^2} \eqno(16a)
$$
and taking the derivative of (14) with respect to $q$ we have
$$
{ v'' \over v} - {v'^2\over v^2} = - {c\over p^2}\, p' \eqno(16b)
$$
Making use of equations (7) and (14) to eliminate $p'$ and $v'$ in
(16b) we find
$$
{v'' \over v} = {c f\over p^3} \eqno(17)
$$
Since $v$, $f$ and $p$ are positive, $v''>0$ and $h = - g' > 0$.
Finally we must show that the integral of $g$ does not diverge. Since
$p$ is continuous and vanishes only at $q=0$ and $q=1$, the only
divergence of the integral of $g$ can occur at these points. We have
chosen $g(1) = 0$ so we must examine the behavior of $g$ at $q=0$.
Near $q=0$, we know $p \sim a_1 q$ so from (14) we have
$$
{v' \over v} \sim {c\over a_1 q} = {\lambda\over q}
$$
which implies that $v\sim q^\lambda$ and
$$
g \sim {1\over \lambda q^{\lambda -1} } \eqno(18)
$$
and therefore
$$
\int_0^1 g(q)\, dq < \infty \qquad {\rm if}\qquad \lambda < 2 \eqno(19)
$$
which is precisely the value of $\lambda$ corresponding to the
asymptotic behavior of the selected front.
Having verified that $g \in \cal D$ we now show that for this choice of
$g$ the equality holds in (13). We must verify that
$$
h p + {f g\over p} = 2\sqrt{fgh}
$$
or equivalently, that
$$
h p = {f g\over p}
$$.
This follows directly from the definition of $g$, $ h p = - g' p =
v'' p/v'^2 $. Using equations (14), (17) and the definition (15) of $g$
the result follows. Having shown that whenever $\lambda < 2$ there
exists a function $g$ for which we obtain the exact speed we have
proven (4).
\section{Example}
In this section we illustrate the results by applying it to the
exactly solvable case $f(u) = u (1 - u) (1 + a u)$ for which it is known
that
$$
c^* = \left\{ \begin{array}{ll}
c = \sqrt{2\over a} + \sqrt{a\over 2}
& \mbox{if $a > 2$} \\
2\sqrt{f'(0)} &\mbox{if $a< 2$}
\end{array} \right.
$$
Let
$$
g(q) = {(1 - q)^{\lambda +1}\over q^{\lambda -1} } \qquad \mbox{with
$\lambda = 1 + {2\over a}$}
$$
then
$$
h = -g' = { (1 - q)^\lambda\over q^\lambda} (\lambda -1) (1 + a x)
$$
and\cite{AStables}
$$
\int_0^1 g(q)\, dq = {\Gamma (\lambda + 2) \Gamma (2-\lambda)\over
\Gamma (4) } \qquad {\rm if}\quad \lambda < 2
$$
We obtain
$$
\int_0^1 \sqrt{f g h}\, dq = \sqrt{\lambda -1 } \left( \int_0^1
{(1-q)^{\lambda -1}\over q^{\lambda -1}} dq + a \int_0^1 (1 -
q)^{\lambda +1} q^{2-\lambda} dq \right)
$$
and therefore
$$
I(g) = 2\sqrt{\lambda -1} \left( 1 + a {\Gamma (3-\lambda)\over
\Gamma (2-\lambda)} {\Gamma (4)\over \Gamma (5)} \right)
$$
Using the definition of $\lambda$ and $\Gamma (z+1) = z \Gamma (z)$
we obtain
$$
I(g) = \sqrt{2\over a} + \sqrt{a\over 2}
$$
which is the exact value for the speed $c^*$.
\section{Conclusion}
We have given a variational characterization of the minimal speed for
which the nonlinear diffusion equation has monotonic fronts. As [AW]
have shown this is the asymptotic speed of propagation of
a sufficiently small positive initial condition $u(x,0)$.
The variational principle we have derived here can also be used to study
the dependence of $c$ on the parameters of $f$. Monotonicity properties
can be inmediatly derived. Derivatives of $c$ with respect to
parameters of $f$ can be obtained easily using the Feynman--Hellmann formula.
\section{Acknowledgments}
This work was partially
supported by Fondecyt project 193-0559.
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\end{document}