\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 113, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/113\hfil Persistence of spreading speed]
{Persistence of spreading speed for the delayed Fisher equation}

\author[S. Pan \hfil EJDE-2012/113\hfilneg]
{Shuxia Pan} 

\address{Shuxia Pan \newline
Department of Applied Mathematics, 
Lanzhou University of Technology,
Lanzhou, Gansu 730050, China}
\email{shxpan@yeah.net}

\thanks{Submitted April 18, 2012. Published July 5, 2012.}
\subjclass[2000]{35C07, 35K57, 37C65}
\keywords{Nonmonotone equation; spreading speed; delayed equation}

\begin{abstract}
 This article concerns  the long time behavior of the delayed Fisher
 equation without quasimonotonicity. When the time delay is small and the
 instantaneous self-limitation effect exists, it is proved that  the spreading
 speed is the same as that of the classical Fisher equation.
\end{abstract}

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

\section{Introduction}

The geographic expansion mode of biological invasion and pathophoresis is one
of the most important topics in population dynamics, and much evidence indicates 
that the mode can be described by the asymptotic spreading. For example, we  
refer to Lewis et al.\ \cite{lewis} and  Murray \cite[Chapter 1]{murray} for the 
spatial spreading of the grey squirrel in the UK. Mathematically, it has been
 proved that a feature of the asymptotic spreading of  some evolutionary models 
can be formulated by  the asymptotic speed of spreading (for short, spreading speed) 
which was first introduced by Aronson and Weinberger \cite{aron1} for the Fisher 
equation. After that, this concept has been widely studied and some important results 
have been established for reaction-diffusion equations, lattice differential 
equations, discrete-time recursions and integral
equations, see Diekmann \cite{diek}, Hsu and Zhao \cite{hsuzhao}, Li et al.\ \cite{llw},  Liang and Zhao \cite{liang}, Lin et al.\ \cite{llr},  Thieme and Zhao \cite{thieme}, Weinberger et al.\ \cite{weinberger}, Zhao \cite{zhao} and the references cited therein.  For the sake of convenience, we first show the following definition.

\begin{definition}\label{d1.1}{\rm
 Assume that $u(x,t)$ is a nonnegative function for
all $x\in \mathbb{R}$, $t>0$. Then $c_{*}>0$ is called the
{\it asymptotic speed of spreading} of $u(x,t)$ if
\begin{itemize}
\item[(a)] $\lim_{t\to \infty}
\sup_{|x|>(c_*+\epsilon)t}u(x,t)=0$ for any given $\epsilon >0$;
\item[(b)] $\liminf_{t\to \infty}
\inf_{|x|<(c_*-\epsilon)t}u(x,t)>0$ for any given $\epsilon \in
(0,c_*)$.
\end{itemize}}
\end{definition}

At the same time,  Berestycki et al.\ in \cite{be} and in \cite{be2} 
presented some examples to illustrate the possible complexity of 
spreading speed of an  unknown function formulated by a scalar equation.

From the viewpoint of monotone dynamical systems, the results mentioned above 
can be applied to an evolutionary system admitting a proper comparison principle 
(see Berestycki et al.\ \cite{be}, Berestycki et al.\ \cite{be2}, Liang and 
Zhao \cite{liang}, Weinberger et al.\ \cite{weinberger}) or controlled by two 
systems generating monotone semiflows (see Hsu and Zhao \cite{hsuzhao}, 
Li et al.\ \cite{llw}).
However,  if an equation does not have a comparison principle near the unstable
 steady state, then these methods fail and the study of asymptotic spreading will 
be very hard. Recently, Lin \cite{lin} considered the spreading speed of the 
following delayed equation
\begin{equation}\label{1}
\frac{\partial u(x,t)}{\partial t}=d \Delta u(x,t)+ru(x,t)
\left[1- u(x,t)-au(x,t-\tau)\right],
\end{equation}
where  $u\in\mathbb{R}$, $x\in\mathbb{R}$, $t>0$, $d>0$, $r>0$, $\tau \ge 0$ and $a>0$ such that the results mentioned above cannot be applied. For $a\in (0,1)$,   $u(x,t)$ defined by \eqref{1} has a spreading speed $2\sqrt{dr}$ if the initial value has nonempty compact support, which is the same as that of the classical Fisher equation (see Lin \cite{lin}). Biologically, $a\in (0,1)$ implies that the instantaneous self-limitation effect  dominates the corresponding delayed effect.

In this article, we shall further consider the asymptotic spreading of \eqref{1}
 if  $a \ge 1$ such that the instantaneous self-limitation effect does not 
dominate the corresponding delayed effect. Using the theory of abstract functional 
differential equations, some properties of the corresponding initial value 
problem of \eqref{1} are proved. In particular,  we obtain a positive constant 
$\tau_0$ such that  for each fixed $c<2 \sqrt{dr}$, we prove that
\[
\liminf_{t\to\infty}\inf_{|x|<ct}u(x,t)>0
\]
when $\tau <\tau_0$ is true and the initial value has nonempty support.
 At the same time, when $c> 2 \sqrt{dr}$ and $\tau\ge 0$, we also confirm that
\[
\limsup_{t\to\infty}\sup_{|x|> ct}u(x,t)=0
\]
if \eqref{1} has an initial value admitting nonempty compact support. 
Therefore, for small delay,  we obtain the persistence of spreading speed 
(if $\tau =0$ holds and the initial value has nonempty compact support, 
then the spreading speed of $u(x,t)$ defined by \eqref{1} is $2\sqrt{dr}$, 
see Lemma \ref{le3}).

\section{Initial Value Problem}

In this section, we present some results on the corresponding initial value 
problem of \eqref{1}. Let
\[
X=\{u(x): u: \mathbb{R}\to\mathbb{R} \text{ is bounded and uniformly continuous}\}.
\]
It is well known that $X$ is a Banach space with respect to the standard 
supremum norm $|\cdot|$. We also denote
\[
X^+= \{u: u\in X\text{ and }u(x)\ge 0\text{ for all }x\in\mathbb{R}\}.
\]
If $a<b$, then
\[
X_{[a,b]}=\{u: a\le u(x)\le b\text{ for all }x\in\mathbb{R}\}.
\]
For each $t>0$, define $T(t):X\to X$ as follows
\[
T(t)u(x)=\frac{1}{\sqrt{4\pi dt}}
\int_{\mathbb{R}}e^{-\frac{(x-y)^2}{4dt}}u(y)dy,\quad u(x)\in X.
\]
Then $T(t):X\to X$ is an analytic semigroup \cite{smithzhao}, and 
$T(t): X^+\to X^+$ is a positive semigroup.
To use the theory of abstract functional differential equations, 
we regard $u(x,t):\mathbb{R}^+\to X$, and $u(t)\in X$ implies that 
$u(x,t)=:(u(t))(x)\in X$. Therefore,
\[
T(t)u(s)=:T(t)u(x,s)=\frac{1}{\sqrt{4\pi dt}}\int_{\mathbb{R}}
e^{-\frac{(x-y)^2}{4dt}}u(y,s)dy,u(s)\in X.
\]

We now consider the following Cauchy  problem
\begin{equation}\label{10}
\begin{gathered}
\frac{\partial u(x,t)}{\partial t}=d \Delta u(x,t)+ru(x,t)
 \left[1- u(x,t)-au(x,t-\tau)\right],\\
u(x,s)=\phi(x,s), \quad x\in\mathbb{R},\; t>0,\; s\in [-\tau, 0],
\end{gathered}
\end{equation}
where all the parameters are the same as those in \eqref{1}.

In Smith and Zhao \cite{smithzhao}, the authors discussed the initial value 
problem of delayed reaction-diffusion equations by the theory in  Martin and 
Smith \cite{m}. Similar to that in \cite[Proof of Theorem 2.2]{smithzhao}, 
we  give the following existence and uniqueness of the mild solution of \eqref{10}.

\begin{lemma}\label{le1}
Assume that $\phi: [-\tau, 0]\to X$ is continuous. Then there exists
 $b\in (0,  \infty]$ such that \eqref{10} has a unique mild solution 
$u:[-\tau, b)\to X$, which can also be formulated by the following integral equation
\begin{equation}\label{2}
u(t)=T(t)\phi(0)+\int_0^t T(t-s) F(u_s)ds
\end{equation}
with
\[
F(u_s)=ru(s)\left[1- u(s)-au(s-\tau)\right],\quad
 u(s-\theta)\in X, \theta\in [0,\tau].
\]
If $t>\tau$, then $u(x,t)$ is a classical solution satisfying \eqref{10}.
 Moreover, if $b$ is bounded, then $|u(t)|\to \infty$ if $t\to b-$.
\end{lemma}

Recently, by the condition of quasipositivity and the positivity of $T(t)$, 
Lin \cite{lin} also proved the following result for any $a\ge 0$.

\begin{lemma}\label{le2}
Assume that $\phi: [-\tau, 0]\to X^+$ is continuous. Then  \eqref{10}
 has a unique mild solution $u:[-\tau, \infty)\to X$ satisfying
\[
u(t)\in X^+\text{ for each }t\in [-\tau, \infty).
\]
Moreover, if $\phi: [-\tau, 0]\to X_{[0,1]}$, then
\[
u(t)\in X_{[0,1]}\text{ for each }t\in [-\tau, \infty).
\]
\end{lemma}

Note that the proof of Wang et al.\ \cite[Proposition 4.3]{wang} only depends 
on the boundedness of solutions, so we have the following conclusion 
by Lemma \ref{le2}.

\begin{lemma}\label{le4}
Assume that $\phi: [-\tau, 0]\to X_{[0,1]}$ is continuous. Then there exists 
a constant $C$ independent of $\tau$ such that
$|\frac{\partial u(x,t)}{\partial t}|<C$ and 
$ |\frac{\partial u (x,t)}{\partial x}|<C$ for all $t\ge 3\tau +2$.
\end{lemma}

Before ending this section, we state some results on the classical Fisher 
equation (we refer to Aronson and Weinberger \cite{aron1}, 
Smith and Zhao \cite{smithzhao}, Ye and Li \cite{yeli}).

\begin{lemma}\label{le3}
Assume that $d_1,r_1$ and $M$ are positive constants. 
Consider the  initial value problem
\begin{equation}
\begin{gathered}
\dfrac{\partial w(x,t)}{\partial t}= d_1\Delta w(x,t)+r_1w(x,t)\left[
1-M w(x,t)\right] ,\quad x\in\mathbb{R},\; t>0, \\
w(x,0)=\varphi(x)\in X^+ , \quad x\in \mathbb{R}. 
\end{gathered} \label{2.8}
\end{equation}
\begin{itemize}
\item[(A1)] For all $t>0$, \eqref{2.8} has a classical solution 
$w(\cdot,t)\in X^+$. If $\varphi (x)$ admits nonempty support, then $w(x,t)>0$ 
for all $x\in\mathbb{R}, t>0$.

\item[(A2)] If $z(\cdot,t)\in X^+$ with $t\in (0,b)$ satisfies
    \begin{equation*}
\begin{gathered}
\dfrac{\partial z(x,t)}{\partial t}\ge (\le) d_1\Delta z(x,t)+r_1z(x,t)\left[
1-M z(x,t)\right] ,\\
z(x,0)\ge (\le)\varphi(x), \end{gathered}
\end{equation*}
then $z(x,t)\ge (\le) w(x,t)$ for all $x\in\mathbb{R},t\in (0,b)$.

\item[(A3)] Assume that $z(t)\in X^+$ with $t\in [0,b)$ satisfies
\[
z(t)\ge (\le) T(t-s)z(s)+\int_s^t T(t-\theta)[r_1z(\theta)(1-Mz(\theta))]d\theta
\]
for all $0\le s\le t<b$. Then $z(x,t)\ge (\le) w(x,t)$ for all
 $x\in\mathbb{R},t\in[0,b)$.

\item[(A4)]  If $\varphi(x)$ admits nonempty support, then
\[
\liminf_{t\to\infty}\inf_{|c|<ct}w(x,t)=\limsup_{t\to\infty}\sup_{|c|<ct}w(x,t)=1/M
\]
for each $c<2\sqrt{d_1r_1}$. Moreover, if $\varphi(x)$ admits nonempty 
compact support, then 
$$
\limsup_{t\to\infty}\sup_{|x|>ct}w(x,t)=0
$$ 
for each $c>2\sqrt{d_1r_1}$.
\end{itemize}
\end{lemma}

\begin{remark}\label{re}{\rm
(A2)--(A3) remain true if $M=0$ or $r_1<0.$}
\end{remark}


\section{Main Results}

\begin{theorem}\label{th1}
Assume that $u(x,t)$ is the mild solution of \eqref{10} and $\phi(0)\in X^+$ 
admits nonempty support. If $\tau >0$ such that $aC\tau<1$, then
\[
\liminf_{t\to\infty}\inf_{|x|<ct}u(x,t)\ge \frac{1-aC\tau}{1+a}
\]
for each $c<2 \sqrt{dr(1-aC\tau)}$, hereafter $C$ is given by Lemma \ref{le4}.
\end{theorem}

\begin{proof}
If $t\ge 4\tau+3$, then Lemma \ref{le4} implies that
\[
|u(x,t)-u(x,t-\tau)|<C\tau.
\]
Thus, we obtain
\begin{equation}\label{ge}
\frac{\partial u(x,t)}{\partial t}\ge d \Delta u(x,t)+ru(x,t)
\left[1- (1+a)u(x,t)-aC\tau \right].
\end{equation}
On the other hand, Lemma \ref{le2} and the positivity of semigroup indicate that
\begin{equation}
\begin{aligned}
u(t) &=T(t-s)u(s)+\int_{s}^{t}T(t-\theta )\left\{ ru(\theta )[1-u(\theta
)-au(\theta -\tau )]\right\} d\theta   \\
&\geq T(t-s)u(s)+\int_{s}^{t}T(t-\theta )[-rau(\theta )]d\theta
\end{aligned} \label{**}
\end{equation}
for any $0\le s\le t <\infty$.
Let $v(x,t)$ be defined by
\begin{equation}
\begin{gathered}
\dfrac{\partial v(x,t)}{\partial t}= d\Delta v(x,t)- rav(x,t) ,\\
v(x,0)=\varphi(x)\in X_{[0,1]}.
\end{gathered} \label{2.8***}
\end{equation}
By Remark \ref{re}, we see that
\[
v(x,t)>0,\quad x\in\mathbb{R},\; t>0
\]
if $\varphi(x)$ admits nonempty compact support.
From Lemma \ref{le3}, we also have
\[
u(x,t)> v(x,t)>0
\]
for any $t>0$. Therefore, $u(x,4\tau +3)>0$ admits nonempty support.
Again by Lemma \ref{le3}, the result is evident  by \eqref{ge}.
The proof is complete.
\end{proof}


From the above proof and Lemma \ref{le3}, we also have the following result.

\begin{corollary}\label{co1}
Let $u(x,t)$ be the mild solution of \eqref{10} and $\delta_1 >0, \delta_2>0$ 
be constants such that
\[
u(x,0) > \delta_1, |x|\le \delta_2.
\]
Then for any $\epsilon <\frac{1-aC\tau}{1+a}$, there exists 
$T=T(\epsilon, \delta_1, \delta_2)$ such that
\[
u(0, t)> \epsilon, t> T.
\]
\end{corollary}

\begin{theorem}\label{th3}
Assume that $\phi(0)\in X^+$ admits nonempty support and $\tau <\tau _0$. 
If  $u(x,t)$ is the mild solution of \eqref{10}, then
\[
\liminf_{t\to\infty}\inf_{|x|< ct}u(x,t) \ge \frac{1-aC\tau}{1+a}
\]
for each fixed $c<2 \sqrt{dr}$.
\end{theorem}

\begin{proof}
In the subsequent proof, we assume that $c<2 \sqrt{dr}$ is a fixed constant. 
It suffices to consider $t\ge 4\tau +3$ such that $u(x,t)$ satisfies \eqref{10} and
$$
\big| \frac{\partial u(x, t-\tau)}{\partial t}\big| <C, \quad
\big| \frac{\partial u(x, t-\tau)}{\partial x}\big| <C.
$$
Since $c<2 \sqrt{dr}$, there exists $\epsilon ' >0$ such that
\[
8\sqrt{dr(1-a \epsilon')} =c +6 \sqrt{dr}.
\]
If $u(x,t-\tau) \le \epsilon '$, then
\begin{equation}\label{xiao}
\frac{\partial u(x,t)}{\partial t}\ge d \Delta u(x,t)+ru(x,t)\left[1- u(x,t)
-a \epsilon '\right].
\end{equation}

If $u(x,t-\tau) > \epsilon '$, then Lemma \ref{le4} and Corollary \ref{co1}
 indicate that there exists $T_1>0$ such that
\[
u(x, t-\tau +T') >\frac{1-aC \tau}{2(1+a)} \text{ for any } T' >T_1.
\]
Note that $u(x,s), s\in [t-\tau, t]$ also satisfies \eqref{**}, then Lemma \ref{le3}
 leads to
\begin{equation}\label{f}
u(x,t)> \delta \epsilon '
\end{equation}
with some fixed $\delta >0$. In particular, the comparison principle 
(Lemma \ref{le3}) also confirms that both $\delta$ and $T_1$ are uniform 
for all $x,t$ satisfying  
$$
u(x,t-\tau) > \epsilon ', t\ge  4\tau +3.
$$
From \eqref{f} and $\epsilon ' <u(x,t-\tau )\le 1$, it is evident that
\begin{equation}\label{da}
\frac{u(x,t-\tau)}{u(x,t)}< \frac{1}{\delta \epsilon '}, t>4\tau +3.
\end{equation}
Combining \eqref{xiao} with \eqref{da}, we obtain
\[
\frac{\partial u(x,t)}{\partial t}\ge d \Delta u(x,t)+ru(x,t)
\left[1 -a \epsilon '-Mu(x,t)\right]
\]
with $M= (1+a/ (\delta \epsilon ')),x\in\mathbb{R}, t> 4\tau +3$.

Using Lemma \ref{le3}, there exists $T_2>0$ such that
\begin{equation}\label{dada}
\inf_{|x|\le c_1 t} u(x,t) > \frac{1 -a \epsilon '}{2M}, \quad
t> T_2, \quad 2c_1= c+2\sqrt{dr(1-a \epsilon')}.
\end{equation}
From Theorem \ref{th1} and $c<c_1$, we further have
\begin{equation}\label{***}
\liminf_{t\to\infty }\inf_{|x|\le c t} u(x,t) \ge \frac{1- aC \tau}{1+a}  .
\end{equation}
In fact,  $u(x,t)$ also satisfies
\[
\frac{\partial u(x,t)}{\partial t}\ge d \Delta u(x,t)+ru(x,t)
\left[1- (1+a)u(x,t)-a C \tau\right],
\]
then for any $\varepsilon \in (0,1)$, \eqref{dada} implies that there exists
 $T_3 >0$ such that
\[
\inf_{|x|\le c_1 t} u(x,t+T_4) \ge \frac{\varepsilon (1- aC \tau)}{1+a}
\]
for any $t>T_2$ and $ T_4 > T_3$. Moreover, if $|x|<c_1 t$ with large $t$, 
then there exists $s=c_1t/c$ such that $|x|<cs$ and $t+T_3 <s$, 
which further implies that
\[
\inf_{|x|\le cs} u(x,s) \ge \frac{\varepsilon (1- aC \tau)}{1+a}.
\]
Furthermore, $t\to \infty$ if and only if $s\to \infty$, and we have
\[
\liminf_{s\to \infty }\inf_{|x|\le cs} u(x,s) 
\ge \frac{\varepsilon (1- aC \tau)}{1+a}.
\]
By the arbitrary nature of $\varepsilon$, \eqref{***} holds and the proof is complete.
\end{proof}


\begin{theorem}\label{th2}
Assume that $\phi(0)\in X^+$ admits nonempty compact support. Then
\[
\limsup_{t\to\infty}\sup_{|x|>ct}u(x,t)=0
\]
for each $c>2 \sqrt{dr}$.
\end{theorem}

\begin{proof}
Lemma \ref{le2} and the positivity of semigroup indicate that
\[
u(t)\le T(t-s)u(s)+\int_s^t T(t-\theta)[ru(\theta)[1- u(\theta)]]d\theta
\]
for any $0\le s\le t <\infty$. By Lemma \ref{le3}, the result is clear.
\end{proof}

Before ending this article, we make the following remark.

\begin{remark}{\rm
In the proof of this paper and that of Lin \cite{lin}, the existence
 of the instantaneous self-limitation effect plays an important role.
 If the effect does not exist, then the discussion will be very difficult 
even if the time delay is small enough. }
\end{remark}

\subsection*{Acknowledgements}
The author would like to thank the referee for his/her careful reading and 
useful comments. The work was supported by the Development Program for 
Outstanding Young Teachers in Lanzhou University of Technology (1010ZCX019).

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\end{document}