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\AtBeginDocument{{\noindent\small
2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal,\\
\emph{Electronic Journal of Differential Equations},
Conference 22 (2015),  pp. 47--51.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document} \setcounter{page}{47}
\title[\hfilneg EJDE-2015/Conf/22 \hfil Competition system with slow diffusion]
{A convergence theorem for a two-species competition system with slow diffusion}

\author[G. Hetzer, L. Tello \hfil EJDE-2015/conf/22 \hfilneg]
{Georg Hetzer, Lourdes Tello}

\dedicatory{Dedicated to Professor Alfonso Casal on  his 70th birthday}

\address{Georg Hetzer \newline
 Department of Mathematics and Statistics \\
  Auburn University \\
 Auburn, AL 36849, USA}
\email{hetzege@auburn.edu}

\address{Lourdes Tello \newline
Department of Applied Mathematics \\
ETS Arquitectura.  Universidad Polit\'ecnica de Madrid \\
 28040 Madrid, Spain}
\email{l.tello@upm.es}

\thanks{Published November 20, 2015.}
\subjclass[2010]{35K57, 35K65}
\keywords{Two-species competition-diffusion system; slow dispersal;
\hfill\break\indent
identical species; convergence to equilibria}

\begin{abstract}
 This article concerns the effect of slow diffusion in
 two-species competition-diffusion problem with
 spatially homogeneous nearly identical reaction terms.
 In this case all (nonnegative) equilibria are spatially homogeneous,
 and the set of nontrivial equilibria is the graph of a $C^1$-curve.
 This article shows convergence of positive solutions to an equilibria
 which is determined by the initial data. The proof relies on the existence
 of a Lyapunov function and is adapted from \cite{EH} which dealt with linear
 diffusion.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

We study the asymptotic behavior of positive solutions of the two-species system
\begin{equation} \label{MainEq}
\begin{gathered}
u_t-\Delta_p u=ug(u,v)\quad \text{in }(0,\infty)\times\Omega,\\
v_t-d\Delta_q v=rvg(u,v)\quad \text{in }(0,\infty)\times\Omega,\\
\partial_{\mathfrak{n}}u=0=\partial_{\mathfrak{n}}v \quad \text{on }
(0,\infty)\times\partial\Omega.
\\
u(0,x)=u_0(x)\quad\quad \quad\quad \mathrm{in }\quad \Omega,\\
v(0,x)=v_0(x)\quad\quad\quad\quad \mathrm{in }\quad \Omega .
\end{gathered}
\end{equation}
under the following hypotheses:
\begin{itemize}
\item[(H1)] $N\in\mathbb{N}$, $\Omega\subseteq\mathbb{R}^N$
bounded smooth domain, $p,q>\max\{2,N\}$, $d,r>0$;

\item[(H2)] $g\in C^2(\mathbb{R}_+^2, \mathbb{R})$,
$\partial_j g<0$ for $j=1,2$, $g(0,0)>0$;
$g$ is negative outside a bounded region.
\end{itemize}

Slow diffusion ($p,q>2$) arises in filtration, and \eqref{MainEq} could,
 e.g., model the spread of microorganisms in lymph nodes.
The case considered here can be thought of as competition between
a species and one of its mutants. The crucial difference is the dispersal,
but no spatial adaptation has taken place, and the fitness function differ
at most by a constant factor $r>0$, in applications $r=1$.

Our main result states that every positive solution of
\eqref{MainEq} converges to one of the nontrivial equilibria. The
same result has been obtained in \cite{EH} for linear diffusion
$p=q=2$. The case where spatial adaption has occurred is different
(isolated equilibria) and has found quite some interest over the
years. The reader is referred to the classical papers \cite{Has} and
\cite{DHMP} for linear dispersal, where the ``slower diffuser''
persists and can invade, to \cite{HMMV} for nonlocal diffusion, and
to \cite{KLS} for linear vs. nonlocal dispersal. Other related
papers are \cite{HNS}, \cite{HLM}, \cite{HLMP}, \cite{HMP}, and
\cite{LMP} but slow dispersal has not been considered to our
knowledge.

We remark that system \eqref{MainEq} with degenerate operators
($p>2$, $q>2$) may involve a time dependent free boundary problem
for some initial data (see e.g \cite{ADS}), but this issue is
not considered here.

The paper is organized as follows. The next section recalls some well-known
results for the solution semiflow of \eqref{MainEq} and outlines the proof
for global existence and nonnegativity. These results are used in Section 3
to establish the convergence result.

\section{Preliminaries}

Let $r\in \{p,q\}$ and $A_r:L^2(\Omega)\supset\operatorname{dom}(A_r)\to L^2(\Omega)$
be the subdifferential of
$$
w\mapsto \begin{cases} \frac{1}{r}\int_\Omega | \nabla w|^r  &
w\in W^{1,r}(\Omega)\\
\infty & w\in L^2(\Omega)\setminus W^{1,r}(\Omega)
\end{cases},
$$
then $A=(A_p,d A_q)$ is the realization of the principal (elliptic)
part of \eqref{MainEq} in $L^2(\Omega)$.
$A$ is densely defined, m-accretive, and generates a completely continuous solution
semigroup in $L^2(\Omega)\times L^2(\Omega)$. Therefore the standard theory
of local Lipschitz perturbations of $A$ (cf. \cite{Vr}, e.g.)
guarantees that \eqref{MainEq} generates a local solution semiflow in
$L^2(\Omega)\times L^2(\Omega)$, if $g$ is smoothly extended to $\mathbb{R}^2$.
The reader is also referred to \cite{SG} for a general setting involving
set-valued solution semiflows.
Since one is dealing with a competition problem, one is interested in
nonnegative solutions only.
Thus, solutions of \eqref{MainEq} satisfying $u(0,\cdot)\ge 0$,
$v(0,\cdot)\ge 0$ should be global and nonnegative.
For our main result it suffices to consider smooth initial
conditions (regularity properties), hence one can assume for solutions
$(u,v)$ of \eqref{MainEq} on $[0,T]$
that $u\in L^p([0,T],W^{1,p}(\Omega))\cap W^{1,p'}([0,T],W^{-1,p'}(\Omega))$
and $v\in L^p([0,T],W^{1,p}(\Omega))\cap W^{1,p'}([0,T],W^{-1,p'}(\Omega))$.
Moreover, if $p,\ q>N$ as assumed in (H1), one has
$W^{1,r}(\Omega)\hookrightarrow C(\overline{\Omega})$ for $r\in\{p,q\}$.

\begin{lemma}\label{GE}
Let {\rm (H1)--(H2)} be satisfied, $T>0$, $u_0,v_0\in \operatorname{dom}(A)$ be
positive, and $(u,v)$ be the solution of \eqref{MainEq} on $[0,T)$
with $(u(0,\cdot),v(0,\cdot))=(u_0,v_0)$. Then $0\le u(t,\cdot)\le
\| u_0\|_\infty +\beta+1$ and $0\le v(t,\cdot)\le \| v_0\|_\infty+\beta+1$
for $0\le t<T$.
\end{lemma}

\begin{proof}
It suffices to deal with the statements for $u$.
Let $\sigma>g(0,0)$ and $w(t,\cdot)=e^{-\sigma t}u(t,\cdot)\ge 0$ for
$t\in [0,T]$. Then $w$ satisfies
\begin{equation}\label{Eq2}
\partial_t w-e^{(p-2)\sigma t}\Delta_p w+\sigma w-wg(e^{\sigma t}w,v)=0.
\end{equation}

Note that the function $h(t,x,y):=\sigma y- yg(e^{\sigma t}y,v(t,\cdot))$
is strictly increasing in $y$ in view of (H2) and
the choice of $\sigma$. Weak $p-$Laplacian comparison theorems have
been established beginning with \cite{CG}, and the proof of proposition 2.2 in
\cite{DT} or that of Lemma 4.9 in \cite{PTT} (Dirichlet case) apply
immediately. We also refer to \cite{DT94} for a more general quasilinear operator.
In fact, let $\phi$ be the solution of $\dot\phi=\phi
g(\phi,0)$, $\phi(0)=0$, and $\psi(t)=e^{-\sigma t}\varphi(t)$, then
$\psi$ satisfies
$$
\dot\psi(t)+\sigma\psi(t)-\psi g(e^{\sigma t}\psi,0)=0,
$$
hence
$$
\dot\psi(t)+\sigma\psi(t)-\psi g(e^{\sigma t}\psi,v(t))\ge 0
$$
in view of (H2). Thus,
\begin{align*}
&\int_\Omega\Bigl( (\partial_t
w(t,x)-\dot\psi(t))[w(t,x)-\psi(t)]^+ -e^{(p-2)\sigma t}\Delta_p
w(t,x)[w(t,x)-\psi(t)]^+\\
&+\sigma(w(t,x)-\psi(t))[w(t,x)-\psi(t)]^+ \\
&-(w(t,x)g(e^{\sigma t}w(t,x), v(t,x))-\psi g(e^{\sigma
t}\psi,v(t,x)))[w(t,x)-\psi(t)]^+\Bigr)\,dx\le 0,
\end{align*}
hence the
m-accretiveness of the $p$-Laplacian and $h$ monotone increasing
imply that $$\int_\Omega [w(t,x)-\psi(t)]^+\,dx \le
\int_\Omega [w(0,x)-\psi(0)]^+\,dx =0,$$ hence $w(t,\cdot)\le
\psi(t)$ for $t\in [0,T)$, therefore, $$u(t,\cdot)\le \phi(t)\le
\| u_0\|_\infty+\beta+1$$ for $t\in [0,T)$. The
nonnegativity of $u$ follows from the same weak comparison argument
and the fact that the constant $0$ solves \eqref{Eq2}.
\end{proof}

\section{Main result}

Let $Z:=\{(y,z)\in\mathbb{R}_+^2: y^2+z^2>0, g(y,z)=0\}$. It follows
from (H2) that there exists a $\beta>0$ with $g(\beta,0)=0$  and a
strictly decreasing function $\gamma\in C^1([0,\beta],\mathbb{R}_+)$
with $Z=\{(y,\gamma(y)):y\in [0,\beta]\}$. In fact,
$\gamma'(y)=-\frac{\partial_1g(y,\gamma(y))}{\partial_2g(y,\gamma(y))}$
and in particular $\gamma'(\beta)<0$.


\begin{theorem}\label{MainTh}
Let {\rm (H1)--(H2)} be satisfied, $u_0,v_0\in \operatorname{dom}(A)$ be positive,
and $(u,v)$ be the solution of \eqref{MainEq} with
$(u(0,\cdot),v(0,\cdot))=(u_0,v_0)$. Then $(u(t,\cdot),v(t,\cdot))$
converges uniformly to some $(\zeta,\eta)\in Z$.
\end{theorem}

\begin{proof}
Select $\epsilon\in (0,\beta)$ with $\epsilon<\min
(\{\gamma(\xi)-\xi\gamma'(\xi):0\le\xi\le\beta\})$.
Set 
$$
\kappa(\xi):=\begin{cases}
\epsilon-\gamma(\xi) & 0\le\xi\le\beta\\
\epsilon/\xi/\beta & \xi>\beta
\end{cases}
$$
and
$$
V_0(y,z):=\int_1^y \big(\frac{\kappa(\xi)}{\xi}\big)\,d\xi +
\frac{1}{r}\int_1^z \big(1-\frac{\epsilon}{\xi}\big)\,d\xi
$$
for $(y,z)\in (0,\infty)\times (0,\infty) $.


Clearly, $V_0$ is continuously differentiable on $\mathbb{R}_+^2$ and has,
as outlined in \cite{EH}, the following properties:
\begin{itemize}
\item $V_0\ge 0$;
\item $V_0(y,z)\to\infty$, if $y\to 0+$ or $z\to 0+$;
\item $V_0(y,z)\to\infty$, if $y\to \infty$ or $z\to \infty$;
\item $\nabla V_0(y,z)\cdot(yg(y,z),rzg(y,z))\le 0$ and ``='' implies
$(y,z)\in Z$.
\end{itemize}
The last statement follows from
$$
\nabla V_0(y,z)\cdot(yg(y,z),rzg(y,z))
=\begin{cases} 
(\epsilon-\gamma(y)) g(y,z)+(1-\frac{\epsilon}{z})zg(y,z) 
& 0<y\le \beta \\
\frac{\epsilon}{\beta}yg(y,z)+(1-\frac{\epsilon}{z})zg(y,z) &
y>\beta .
\end{cases}
$$
Thus, we obtain  $-\gamma(y)g(y,z)+zg(y,z)=(z-\gamma(y))g(y,z)$ 
if $0<y<\beta$, which is $\le 0$, since 
$\mathrm{sgn}(z-\gamma(y))=-\mathrm{sgn}(g(x,y))$.
 Note that the expression is equal to 0, if and only if $(y,z)\in Z$.  
If $y\ge\beta$ and $z>0$, then  $\frac{\epsilon}{\beta}y-\epsilon>0$
 and $g(y,z)<0$, hence
$\frac{\epsilon}{\beta}yg(y,z)+(1-\frac{\epsilon}{z})zg(y,z)<0 $.

Set 
$$
V(\varphi,\psi):=\int_\Omega V_0(\varphi(x),\psi(x))\,dx 
\quad \text{for }\varphi,\psi\in L^\infty(\Omega).
$$ 
Then
\begin{align*}
&\frac{d}{dt} V(u,v)(t) \\
&=\int_\Omega \partial_1
V_0(u(t,x),v(t,x))u_t(t,x)\,dx+ \int_\Omega \partial_2
V_0(u(t,x),v(t,x))v_t(t,x)\,dx\\
&=\int_\Omega \partial_1 V_0(u(t,x),v(t,x))\Delta_p u(t,x)\,dx
+ d\int_\Omega \partial_1 V_0(u(t,x),v(t,x))\Delta_q u(t,x)\,dx \\
&\quad +\int_\Omega\nabla V_0(u(t,x),v(t,x))
\cdot(u(t,x)g(u(t,x),v(t,x)),rv(t,x)g(u(t,x),v(t,x))\,dx.
\end{align*}
Integration by parts shows that
$$
\int_\Omega h(w(x)) \Delta_p w(x)= -\int_\Omega
h'(w(x))|\nabla w(x)|^p\le 0,
$$
if $h\in C^1(\mathbb{R})$ is nondecreasing and 
$w\in\operatorname{dom}(A_p)$. This
and the corresponding $A_q$ statement imply 
\begin{gather*}
\int_\Omega \partial_1 V_0(u(t,x),v(t,x))\Delta_p u(t,x)\,dx\le 0,
\\
d\int_\Omega \partial_1 V_0(u(t,x),v(t,x))\Delta_q u(t,x)\,dx\le 0,
\end{gather*}
which yields that
$\frac{d}{dt} V(u,v)(t)\le 0$ and equal to zero, if and only if
$(u,v)(t)\in Z$. Thus, the $\omega$-limit set of $(u,v)$ contains
only pairs $(\zeta,\eta)$ with $(\zeta(x),\eta(x))\in Z$ for $x\in
\Omega$. Since the $\omega$-limit set is backward invariant (cf.
\cite{Ha}), each $(\zeta,\eta)$ is constant on $\Omega$.

Assume that $(a_j,\gamma(a_j))\in \omega(u,v)$ for $j=1,2$ and that 
$a_1<a_2$, then $(\rho,\gamma(\rho)) \in \omega(u,v)$, and we can assume 
without loss of generality that $0<a_1<a_2<\beta$.
Moreover, $\rho\mapsto V_0(\rho,\gamma(\rho))$ is constant, hence 
$\frac{\epsilon-\gamma(\rho)}{\rho} +\frac{1}{r}
\bigl(1-\frac{\epsilon}{\gamma(\rho)}\bigr)\gamma'(\rho)=0$ for 
$a_1\le\rho\le a_2$.
This yields $\gamma'(\rho)=\frac{r}{\rho}>0$ for $a_1\le\rho\le a_2$ 
which contradicts $\gamma'<0$.
\end{proof}

\begin{remark} \rm
Our results imply that mutations affecting dispersal alone do not
drive the original species into extinction.
\end{remark}

\subsection*{Acknowledgments}
The work by L. Tello is partially supported by the research projects
MTM2013-42907-P and MTM2014-57113-P of Ministerio de Econom\'ia y
Competitividad, Spain.

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\end{document}