\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 09, pp. 1--6.\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.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/??\hfil Nonexistence of solutions]
{Nonexistence of solutions to KPP-type equations
of dimension greater than or equal to one}
\author[J. Engl\"{a}nder, P. L. Simon\hfil EJDE-2006/??\hfilneg]
{J\'{a}nos Engl\"{a}nder, P\'{e}ter L. Simon} % in alphabetical order
\address{J\'{a}nos Engl\"{a}nder \hfill\break
Department of Statistics and Applied Probability\\
University of California, Santa Barbara\\
CA 93106-3110, USA}
\email{englander@pstat.ucsb.edu}
\urladdr{http://www.pstat.ucsb.edu/faculty/englander}
\address{P\'{e}ter L. Simon \hfill\break
Department of Applied Analysis, E\"otv\"os Lor\'and University\\
P\'{a}zm\'{a}ny P\'{e}ter S\'{e}t\'{a}ny 1/C, H-1117 Budapest,
Hungary}
\email{simonp@cs.elte.hu}
\urladdr{http://www.cs.elte.hu/$\sim$simonp}
\date{}
\thanks{Submitted September 19, 2005. Published January 24, 2006.}
\subjclass[2000]{35J60, 35J65, 60J80}
\keywords{KPP-equation; semilinear elliptic equations;
\hfill\break\indent
positive bounded solutions; branching Brownian-motion.}
\begin{abstract}
In this article, we consider a semilinear elliptic equations of
the form $\Delta u+f(u)=0$, where $f$ is a concave function. We
prove for arbitrary dimensions that there is no solution bounded
in $(0,1)$. The significance of this result in probability theory
is also discussed.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction and statement of main result}
In this article, we study semilinear elliptic equations of the
form $\Delta u+f(u)=0$. On the nonlinear term $f:[0,1] \to \mathbb{R}$
we assume that
\begin{itemize}
\item[(i)] $f$ is continuous,
\item[(ii)] $f$ is positive on $(0,1)$,
\item[(iii)] the mapping $z\mapsto f(z)/z$ is strictly decreasing.
\end{itemize}
Under these three conditions, we consider the Kolmogorov Petrovskii
Piscunov-type (KPP-type) equation
\begin{gather}
\Delta u+f(u)=0 \label{eqn} \\
0__0$. (In fact this particular nonlinearity is intimately
related to the distribution of a \emph{branching Brownian motion};
see more on the subject in the next paragraph.) We will present a
proof for our result that works basically for concave functions; in
fact, (iii) of Assumption 1 is related to the concaveness of the
function.
The connection between the KPP equation and branching Brownian
motion has already been discovered by H. P. McKean --- it first
appeared in the classic work \cite{McK1975,McK1976}.
Let $Z=(Z(t))_{t\geq 0}$ be the $d$-dimensional binary branching
Brownian motion with a spatially and temporally constant branching
rate $\beta>0$. The informal description of this process is as
follows. A single particle starts at the origin, performs a Brownian
motion on $\mathbb{R}^d$, after a mean--$1/\beta$ exponential time
dies and produces two offspring, the two offspring perform
independent Brownian motions from their birth location, die and
produce two offspring after independent mean--$1/\beta$ exponential
times, etc. Think of $Z(t)$ as the subset of $\mathbb{R}^d$
indicating the locations of the particles $z_1^t,\dots ,z^{N_t}_t$
alive at time $t$ (where $N_t$ denote the number of particles at
$t$). Write $P_{x}$ to denote the law of $Z$ when the initial
particle starts at $x$. The natural filtration is denoted by
$\{\mathcal {F}_t,\; t\ge 0\}$.
Then, as is well known (see e.g. \cite[Chapter 1]{D02}), the law
of the process can be described via its Laplace functional as
follows. If $f$ is a positive measurable function, then
\begin{equation}\label{Laplace.func}
E_x\exp \Big(-\sum_{i=1}^{N_t} f(z_i^t)\Big)=1-u(x,t),
\end{equation}
where $u$ solves the initial value problem
\begin{equation} \label{par.eqn}
\begin{gathered}
\dot{u}=\frac{1}{2}\Delta u+f(u) \quad \text{in }
\mathbb{R}^d\times\mathbb{R}_+ \\
u(\cdot,0)=1-e^{-f(\cdot)} \quad \text{in }\mathbb{R}^d \\ %\label{IC}\\
0\le u\le 1 \quad \text{in } \mathbb{R}^d\times\mathbb{R}_+,%\label{same.cond}
\end{gathered}
\end{equation}
with $f$ of the form \eqref{specialNL}.
Equation \eqref{eqn}-\eqref{cond} appears when one studies certain
`natural' martingales associated with branching Brownian motion (see
e.g. \cite{EK}). To understand this, let $\mathcal{\widehat
F}_t:=\sigma(\bigcup_{s\ge t}\mathcal{F}_s)$ and consider the tail
$\sigma$-algebra $\mathcal{\widehat F}_{\infty}:=\bigcap_{t\ge
0}\mathcal{\widehat F}_s$. Choosing appropriate (sequences of) $f$'s
one can then express the probabilities of various events $A_t\in
\mathcal{\widehat F}_t$, for $t>0$, in terms of the function $u$ in
\eqref{par.eqn}. Letting $t\to\infty$ then leads to the conclusion
that if $A\in \mathcal{\widehat F}_{\infty}$ denotes a certain tail
event (e.g. having strictly positive limit for a certain nonnegative
`natural' martingale, or local/global extinction) then the function
$u(x):=P_x(A)$ is either constant ($=0$ or $=1$), or it must solve
\eqref{eqn}-\eqref{cond}. Hence, it immediately follows from our
main theorem that \emph{the tail $\sigma$-algebra is trivial}, that
is, all those events $A$ satisfy $P_{\cdot}(A)\equiv 0$ or
$P_{\cdot}(A)\equiv1$.
Note that if $\beta>0$ is replaced by a smooth nonnegative function
$\beta(\cdot)$ that does not vanish everywhere, then this
corresponds to having \emph{spatially dependent} branching rate for
the branching Brownian motion. It would be desirable therefore to
investigate whether our main theorem can be generalized for such
$\beta$'s.
\section{Proof of the theorem}
The proof is based on two ideas: The application of the semilinear
elliptic maximum principle, which is generalized here fore concave
functions, and a comparison between the semilinear and the linear
problems. Using these two ideas we will show that the \emph{minimal
positive solution} of \eqref{eqn} is $u_{\min}\equiv 1$, hence
\eqref{eqn} has no solution satisfying \eqref{cond}.
First we state and prove a semilinear maximum principle.
The results in this form is a
generalization of \cite[Proposition 7.1]{EP99}
for the particular case when the elliptic
operator is $L=\Delta$.
\begin{lemma}[Semilinear elliptic maximum principle]\label{emp}
Let $f:[0,\infty ) \to \mathbb{R}$ be a continuous function, for
which the mapping $z\mapsto f(z)/z$ is strictly decreasing.
Let $D\subset \mathbb{R}^{d}$ be a bounded domain with smooth boundary.
If $v_{i}\in C^2(D)\cap C(\bar D)$\ satisfy $v_{i}>0$ in $D$,
$\Delta v_{i}+f( v_{i})=0$,
in $D$ for $i=1,2$, and $v_{1}\ge v_{2}$ on $\partial D$, then
$v_{1}\ge v_{2}$ in $\bar D$.
\end{lemma}
\begin{proof}
Note that the function $w:=v_1-v_2$ satisfies
\begin{equation}
\Delta w + f(v_1)-f(v_2)=0 . \label{e3}
\end{equation}
We show that $w\ge 0$ in $D$. Suppose to the contrary that there
exists a point $y\in D$ where $w$ is negative. Let $\Omega_0:=\{x\in
D\mid w(x)<0\}$. Let $\Omega$ be the connected component of
$\Omega_0$ containing $y$. Since $w \geq 0$ on $\partial D$, one has
$\Omega \subset \subset D$ and
\begin{equation}
w<0 \quad \mbox{in } \Omega, \quad w=0 \quad \mbox{on } \partial \Omega .
\label{e4}
\end{equation}
Let us multiply the
equation $\Delta v_{1}+f( v_{1})=0$ by $w$ and equation \eqref{e3}
by $v_1$, then subtract the second equation from the first, and
integrate on $\Omega$. Using that $w=v_1-v_2$ one obtains
\begin{equation}
I+II:=\int_{\Omega} (w \Delta v_1 - v_1 \Delta w) +
\int_{\Omega} (v_1 f(v_2) -v_2 f(v_1)) =0. \label{5}
\end{equation}
Using Green's second identity and that $w=0$ on $\partial \Omega$,
along with the fact that $\partial_{\nu} w \geq 0$ on $\partial
\Omega$, we obtain
$$
I= -\int_{\partial \Omega} v_1
\partial_{\nu} w \leq 0,
$$
where $\nu$ denotes the unit outward normal to $\partial \Omega$.
Furthermore, since $v_10$
(this is automatically satisfied under assumption that
the mapping $z\mapsto f(z)/z$ is strictly decreasing).
Then for any $y\in \mathbb{R}^d$ and
$p\in (0,1)$ there exists a ball $\Omega:= B_R(y)$ (with some
$R>0$) and a radially symmetric $C^2$ function $v:\Omega \to
\mathbb{R}$ such that
\begin{gather*}
\Delta v + f(v)= 0 \\
v>0 \quad \mbox{in } \Omega\\
v=0 \quad \mbox{on } \partial\Omega\,\quad v(y)=p \,.
\end{gather*}
\end{lemma}
\begin{proof} We show the existence of a radially
symmetric solution of the form $v(x)=V(|x-y|)$. Let
$V\in C^2([0,\infty))$ be the solution of the initial value problem
\begin{equation}
\begin{gathered}
(r^{d-1}V'(r))'+r^{d-1}f(V(r))=0 \\
V(0)=p, \ V'(0)=0 .
\end{gathered}\label{e6}
\end{equation}
Writing $\Delta$ in polar coordinates, one sees that it is
sufficient to prove that there exists an $R>0$ such that $V(R)=0$
and $V(r)>0$ for all $r\in [0,R)$.
To this end, consider the \emph{linear} initial value problem
\begin{equation} \label{e7}
\begin{gathered}
(r^{d-1}W'(r))'+r^{d-1}m W(r)=0 \\
W(0)=p, \ W'(0)=0 ,
\end{gathered}
\end{equation}
where $m>0$ is chosen so that $f(u)>mu$ holds for all $u\in (0,p)$.
(Our assumptions on $f$ guarantee the existence of such an $m$.) It
is known that $W$ has a first root, which we denote by $\rho$. Note
that in this case $-m$ is the first eigenvalue of the Laplacian on
the ball $B_{\rho}$. We now show that $V$ has a root in $(0,\rho]$.
In order to do so let us multiply \eqref{e7} by $V$ and \eqref{e6} by
$W$, then subtract one equation from the other, and finally,
integrate on $[0,\rho]$. We obtain
\begin{equation}
\begin{aligned}
I+II&:=\int_0^{\rho} [(r^{d-1}W'(r))'V(r) -
(r^{d-1}V'(r))'W(r)]\, \mbox{d}r \\
&\quad +\int_0^{\rho} r^{d-1}[mW(r)V(r) - W(r)f(V(r))]\, \mbox{d}r =
0\,.
\end{aligned}\label{e8}
\end{equation}
Suppose now that $V$ has no root in $(0,\rho]$. Then, integrating by
parts, $ I = \rho^{d-1}W'(\rho)V(\rho)<0. $
Next, observe that by integrating \eqref{e6}, one gets $V'(r)<0$
(i.e. $V$ is decreasing). Hence $V(r)__