###### Abstract

In this paper we put forward a viral propagation model with nonlinear
infection rate and free boundaries and investigate the dynamical properties.
This model is composed of two ordinary differential equations and one partial
differential equation, in which the spatial range of the first equation is the
whole space $\mathbb{R}$, and the last two equations have free boundaries. As
a new mathematical model, we prove the existence, uniqueness and uniform
estimates of global solution, and provide the criteria for spreading and
vanishing, and long time behavior of the solution components $u,v,w$.
Comparing with the corresponding ordinary differential systems, the Basic
Reproduction Number ${\cal R}_{0}$ plays a different role. We find that when
${\cal R}_{0}\leq 1$, the virus cannot spread successfully; when ${\cal
R}_{0}>1$, the successful spread of virus depends on the initial value and
varying parameters.

Keywords: viral propagation model, free boundaries, basic reproduction number,
spreading-vanishing, long time behavior.

AMS Subject Classification (2100): 35K57, 35B40, 35R35, 92D30

A viral propagation model with nonlinear
infection rate and free boundaries111This work was supported by NSFC Grants
11771110, 11971128
Lei Lia, Siyu Liub,c, Mingxin Wanga,222Corresponding author. E-mail:
<EMAIL_ADDRESS>
a. School of Mathematics, Harbin Institute of Technology, Harbin 150001, PR
China.

b. School of Science and Engineering, The Chinese University of Hong Kong,
Shenzhen 518172, China.

c. School of Mathematical Sciences, University of Science and Technology of
China, Hefei 230026, China.

##  1 Introduction

Background In order to clarify the pathogenesis of diseases and seek effective
treatment measures, viral dynamics have been a hot research topic (cf. [1,
2]), which usually cannot be answered by biological experimental methods alone
but require the help of mathematical models. For this reason, a simple model
was introduced few decades ago by Nowak and Bangham [3]. See also Nowak and
May [4]. The basic model of viral dynamics is the following set of
differential equations

$\displaystyle\begin{cases}u^{\prime}=\theta-au-buw,\\\ v^{\prime}=buw-cv,\\\
w^{\prime}=kv-qw,\end{cases}$ (1.1)

where $u$, $v$ and $w$ represent the population of uninfected cells, infected
cells and viruses, respectively; uninfected cells are produced at a constant
rate $\theta$ and with death rate $au$; $cv$ is the death rate of infected
cells; virus particles $w$ infect uninfected cells with rate $buw$, and
meanwhile virus particles are produced by infected cells with rate $kv$ and
have death rate $qw$. It has been shown that if the Basic Reproduction Number
${\cal R}_{0}={\theta kb}/(acq)<1$, then the system returns to the uninfected
state $(\theta/a,0,0)$. If ${\cal R}_{0}>1$, then the system will converge to
the unique positive equilibrium state
$\big{(}\frac{qc}{kb},\,\frac{\theta}{c}-\frac{aq}{kb},\,\frac{\theta
k}{qc}-\frac{a}{b}\big{)}$. This indicates that in the initial stage of
infection, if each infected cell infects less than one cell on average, then
the infection cannot spread; if each infected cell infects ${\cal R}_{0}>1$
cells on average, the number of infected cells increases and the number of
uninfected cells declines.

Mathematical Model To investigate the impact of spatial dynamics on this
model, Stancevic et al. [5] extended this model to include spatially random
diffusion and spatially directed chemotaxis. Invoked by their ideas, we give
the basic model assumptions as follows:

(i) A nonlinear infection rate can happen due to saturation at high virus
concentration, where the infectious fraction is so high that exposure is very
likely. Moreover, with the increase of the virus concentration the living
environment for cells becomes worse and worse. Thus, it is reasonable for us
to assume that the rate of infection for virus and the virion production rate
for infected cells are both nonlinear. Here we use

$f_{1}(u,w)=\theta-au-\frac{buw}{1+w},\ \ f_{2}(u,v,w)=\frac{buw}{1+w}-cv,\ \
f_{3}(v,w)=\frac{kv}{1+w}-qw$

instead of the three terms in the right hand side of (1.1).

(ii) We assume that the major spatial dispersal comes from the moving
(diffusion) of viruses in vivo, while both the uninfected and infected cells
are immobile (do not diffuse). So we add only a diffusion term to the
differential equation of viruses;

(iii) Since the infected cells are caused by viruses, their distribution range
is the same;

(iv) The distribution of viruses and infected cells is a local range, which is
small relative to the distribution of uninfected cells, so we think that
uninfected cells are distributed over the whole space. Such kind of
assumptions have been used in the species invasion models (cf. [6, 7, 8] for
example);

(v) Initially, viruses are distributed over a local range $\Omega_{0}$ (the
initial habitat). They will spread from boundary to expand their habitat as a
result of the spatial dispersal freely. That is, as time $t$ increases,
$\Omega_{0}$ will evolve into expanding region $\Omega(t)$ with expanding
front $\partial\Omega(t)$. Initial function $w_{0}(x)$, and as a result
$v_{0}(x)$, will evolve into positive functions $w(t,x)$ and $v(t,x)$ which
vanish on the moving boundary $\partial\Omega(t)$;

(vi) For simplicity, we restrict our problem to the one dimensional case.
Based on the deduction of free boundary conditions given in [9], we have the
following free boundary conditions

$g^{\prime}(t)=-\mu w_{x}(t,g(t)),\ \ \ h^{\prime}(t)=-\beta w_{x}(t,h(t)).$

All of these assumptions (i)-(vi) suggest the following model, which governs
the spatial and temporal evolution of viruses and cells, as well as free
boundaries:

$\begin{cases}u_{t}=f_{1}(u,w),&t>0,\ -\infty<x<\infty,\\\
v_{t}=f_{2}(u,v,w),&t>0,\ g(t)<x<h(t),\\\ w_{t}-dw_{xx}=f_{3}(v,w),&t>0,\
g(t)<x<h(t),\\\ v(t,x)=w(t,x)=0,&t>0,\ x\notin(g(t),h(t)),\\\
g^{\prime}(t)=-\mu w_{x}(t,g(t)),\,\,\,h^{\prime}(t)=-\beta
w_{x}(t,h(t)),&t\geq 0,\\\ u(0,x)=u_{0}(x),&-\infty<x<\infty,\\\
v(0,x)=v_{0}(x),\ \ \ w(0,x)=w_{0}(x),&-h_{0}\leq x\leq h_{0},\\\
h(0)=-g(0)=h_{0},\end{cases}$ (1.2)

where $x=g(t)$ and $x=h(t)$ are the moving boundaries to be determined
together with $u(t,x)$, $v(t,x)$ and $w(t,x)$;
$d,~{}\theta,~{}a,~{}b,~{}c,~{}k,~{}q,~{}\mu,~{}\beta,~{}h_{0}$ are positive
constants.

Denote by $C^{1-}(I)$ the space of Lipschitz continuous functions in $I$. We
assume that the initial functions $u_{0},v_{0},w_{0}$ satisfy

$\left\\{\begin{array}[]{ll}u_{0}\in C^{1-}(\mathbb{R})\cap
L^{\infty}(\mathbb{R}),\ \ v_{0}\in C^{1-}([-h_{0},h_{0}]),\ \ w_{0}\in
W^{2}_{p}((-h_{0},h_{0})),\\\\[1.42262pt] v_{0}(\pm h_{0})=w_{0}(\pm
h_{0})=0,\ \ w^{\prime}(-h_{0})>0,\ \ w^{\prime}(h_{0})<0,\\\\[1.42262pt]
u_{0}>0\ \ \text{in }\,\mathbb{R},\ \ \ v_{0},w_{0}>0\ \
\text{in}\,(-h_{0},h_{0})\end{array}\right.$ (1.3)

with $p>3$. Denote by $L_{0}$ and $L_{*}$ the Lipschitz constant of $u_{0}$
and $v_{0}$, respectively.

Partially degenerate reaction-diffusion systems, which mean that several
diffusion coefficients are zeros, have been increasingly applied to
epidemiology, population biology etc; see [10, 11], for example. Some
researchers have introduced the Stefan type free boundary to the partially
degenerate systems, please refer to [12, 13, 14, 15] and the references
therein.

Aims and Main Results This paper concerns with the dynamics of (1.2). The
global existence, uniqueness, regularity and uniform estimates in time $t$ of
solution are first studied. Then a spreading-vanishing dichotomy is
established, i.e., either

(i) Spreading (virus persistence): the virus successfully infects the
uninfected cells and spreads itself to the uninfected area in the sense that
$\displaystyle\lim_{t\to\infty}h(t)=-\lim_{t\to\infty}g(t)=\infty$, and

$\displaystyle\displaystyle\limsup_{t\to\infty}\|v(t,\cdot)\|_{C([g(t),h(t)])}>0,\
\displaystyle\limsup_{t\to\infty}\|w(t,\cdot)\|_{C([g(t),h(t)])}>0.$

In addition, if $\mathcal{R}_{0}+\sqrt{\mathcal{R}_{0}}>b/a$, where ${\cal
R}_{0}={\theta kb}/(acq)$, then

$\displaystyle\displaystyle\underline{u}_{\infty}\leq\liminf_{t\to\infty}u(t,x)\leq\limsup_{t\to\infty}u(t,x)\leq\bar{u}_{\infty},$
$\displaystyle\displaystyle\underline{v}_{\infty}\leq\liminf_{t\to\infty}v(t,x)\leq\limsup_{t\to\infty}v(t,x)\leq\bar{v}_{\infty},$
$\displaystyle\displaystyle\underline{w}_{\infty}\leq\liminf_{t\to\infty}w(t,x)\leq\limsup_{t\to\infty}w(t,x)\leq\bar{w}_{\infty}$

locally uniformly in $\mathbb{R}$ for some positive constants
$\underline{u}_{\infty}$, $\bar{u}_{\infty}$, $\underline{v}_{\infty}$,
$\bar{v}_{\infty}$, $\underline{w}_{\infty}$ and $\bar{w}_{\infty}$.
Particularly, under a stronger assumption that $b\leq 2a$, we will derive

$\lim_{t\to\infty}u(t,x)=u^{*},\ \ \lim_{t\to\infty}v(t,x)=v^{*},\ \
\lim_{t\to\infty}w(t,x)=w^{*}\ \ {\rm locally~{}uniformly~{}in~{}}\mathbb{R},$

where $(u^{*},v^{*},w^{*})$ is the unique positive root of (4.12);

or

(ii) Vanishing (virus dies out): the virus $w$ and the infected cells $v$ will
vanish in a bounded area, i.e.,
$-\infty<\displaystyle\lim_{t\to\infty}g(t)<\lim_{t\to\infty}h(t)<\infty$ and

$\displaystyle\lim_{t\to\infty}\|v(t,\cdot)\|_{C([g(t),\,h(t)])}=\lim_{t\to\infty}\|w(t,\cdot)\|_{C([g(t),\,h(t)])}=0,\
\ \lim_{t\to\infty}u=\theta/{a}\ \ \mbox{uniformly\, in }\ \mathbb{R}.$

Moreover,
$\displaystyle\lim_{t\to\infty}h(t)-\lim_{t\to\infty}g(t)\leq\pi\sqrt{acd/(\theta
kb-acq)}$ if ${\cal R}_{0}>1$.

As for the Basic Reproduction Number ${\cal R}_{0}=\theta kb/(acq)$, in our
results, we will show that it plays a different role, comparing with the
corresponding ordinary differential systems. When ${\cal R}_{0}\leq 1$,
vanishing always happens, that is, the virus cannot spread successfully. On
the other hand, when ${\cal R}_{0}>1$, we have a criterion as follows: if the
initial occupying area $[-h_{0},h_{0}]$ is beyond a critical size, namely
$2h_{0}\geq\pi\sqrt{acd/(\theta kb-acq)}$, then spreading happens regardless
of the moving parameter $\mu$, $\beta$ and initial population density
$(u_{0},v_{0},w_{0})$. While $2h_{0}<\pi\sqrt{acd/(\theta kb-acq)}$, whether
spreading or vanishing happens depends on the initial population density
$(v_{0},w_{0})$ and the moving parameter $\mu$ and $\beta$.

The paper is organized as follows. Section 2 concerns with the existence,
uniqueness and uniform estimates of global solution. In Section 3 we give some
preliminaries which will be used later. In Section 4 we study the long time
behavior of solution components $u,v,w$, and in Section 5 we discuss the
criteria for spreading and vanishing. At the last section, we give a brief
discussion.

Before ending this section we mention that in recent years, more and more free
boundary problems of reaction diffusion systems have been introduced to
describe the dynamics of species after the pioneering work [17]. Interested
readers can refer to, except for the above cited papers, [18, 19, 20, 21, 22]
for competition models, [23, 24, 25] for prey-predator models.

##  2 Existence, uniqueness and uniform estimates of solution of (1.2)

In this section we prove the global existence and uniqueness of the solution
to problem (1.2). For convenience, we first introduce some notations. Denote

$A_{1}=\max\left\\{\|u_{0}\|_{\infty},\,\theta/a\right\\},\ \
B_{1}=\|v_{0}\|_{\infty}+1,\ \ B_{2}=\|w_{0}\|_{\infty}+1,$

$\mathcal{A}=\\{a,b,c,d,k,q,h_{0},\mu,\beta,\alpha,A_{1},B_{1},B_{2},\|w_{0}\|_{W^{2}_{p}((-h_{0},h_{0}))},w^{\prime}_{0}(\pm
h_{0})\\},$

$\Pi_{T}=[0,T]\times\mathbb{R},\,\ \ \Delta_{T}=[0,T]\times[-1,1],\,\ \
D^{T}_{g,h}=\left\\{0\leq t\leq T,\ g(t)<x<h(t)\right\\}$.

Let $X$ be a Banach space and $\varphi,\psi\in X$. Denote
$\|\varphi,\,\psi\|_{X}=\max\\{\|\varphi\|_{X},\,\|\psi\|_{X}\\}$ for
simplicity.

######  Theorem 2.1.

(Local solution) For any given $\alpha\in(0,1)$ and $p>3/(1-\alpha)$, there
exists a $T>0$ such that the problem (1.2) has a unique local solution
$(u,v,w,g,h)\in C^{1,1-}(\Pi_{T})\times C^{1,1-}(\overline{D}^{T}_{g,h})\times
W^{1,2}_{p}(D^{T}_{g,h})\times[C^{1+\frac{\alpha}{2}}([0,T])]^{2}$. Moreover,

$\displaystyle u>0\ \ {\rm in}\ \ \Pi_{T};\ \ v,w>0\ \ {\rm in}\ \
D^{T}_{g,h};\ \ g^{\prime}(t)<0,\ \ h^{\prime}(t)>0\ \ {\rm in}\ \ [0,T],$

where $u\in C^{1,1-}(\overline{D}^{T}_{g,h})$ means that $u$ is continuously
differentiable in $t\in[0,T]$ and is Lipschitz continuous in $x\in[g(t),h(t)]$
for all $t\in[0,T]$.

###### Proof.

Invoked by the proof of [13, Theorem 2.1] and [14, Theorem 1.1], we divide the
proof into several steps. Unless otherwise specified in the proof, positive
constants $C_{i}$ depend only on $\mathcal{A}$.

Step 1: Given $T>0$, we say $u\in C^{1-}_{x}(\Pi_{T})$ if there is a constant
$L_{u}(T)$ such that

$|u(t,x_{1})-u(t,x_{2})|\leq L_{u}(T)|x_{1}-x_{2}|,\ \ \forall\
x_{1},x_{2}\in\mathbb{R},\,\,0<t\leq T.$

For $s>0$, define

$\mathbb{X}_{u_{0}}^{s}=\left\\{\phi\in C(\Pi_{s}):\phi(0,x)=u_{0}(x),\
0\leq\phi\leq A_{1}\right\\}.$

For any given $u\in\mathbb{X}_{u_{0}}^{1}\cap C^{1-}_{x}(\Pi_{1})$ we consider
the following problem

$\displaystyle\begin{cases}v_{t}=f_{2}(u(t,x),v,w),&t>0,\ g(t)<x<h(t),\\\
w_{t}-dw_{xx}=f_{3}(v,w),&t>0,\ g(t)<x<h(t),\\\ v(t,x)=w(t,x)=0,&t>0,\
x\notin(g(t),h(t)),\\\ g^{\prime}(t)=-\mu w_{x}(t,g(t)),\,h^{\prime}(t)=-\beta
w_{x}(t,h(t))&t\geq 0,\\\ v(0,x)=v_{0}(x),w(0,x)=w_{0}(x),&|x|\leq h_{0},\\\
h(0)=-g(0)=h_{0}.\end{cases}$ (2.1)

By [14, Theorem 1.1], we know that for some $0<T\ll 1$, (2.1) has a unique
solution $(v,w,g,h)\in C^{1,1-}(\overline{D}^{T}_{g,h})\times
C^{\frac{1+\alpha}{2},1+\alpha}(\overline{D}^{T}_{g,h})\times[C^{1+\frac{\alpha}{2}}([0,T])]^{2}$.
Moreover,

$\displaystyle\begin{cases}-w_{x}(t,h(t)),\ w_{x}(t,g(t))>0\ \ \mbox{in}\ \
[0,T];\ \ 0<v\leq B_{1},\ 0<w\leq B_{2}\ \ \mbox{in}\ \ D^{T}_{g,h},\\\
\|w\|_{W^{1,2}_{p}(D^{T}_{g,h})}+\|w_{x}\|_{C(\overline{D}^{T}_{g,h})}+\|g,h\|_{C^{1+\frac{\alpha}{2}}([0,T])}\leq
M,\end{cases}$ (2.2)

where $M$ depends only on $\mathcal{A}$.

Step 2: For the function $w(t,x)$ obtained in Step 1, we consider the
following parameterized ODE problem,

$\displaystyle\left\\{\begin{aligned}
&\tilde{u}_{t}=f_{1}(\tilde{u},w(t,x)),&&(t,x)\in(0,T]\times\mathbb{R},\\\
&\tilde{u}(0;x)=u_{0}(x)>0,&&x\in\mathbb{R}.\end{aligned}\right.$ (2.3)

By the standard ODE theory, (2.3) has a unique solution $\tilde{u}\in
C^{1,1-}(\Pi_{T})$ and $0<\tilde{u}\leq A_{1}$.

Now we estimate the Lipschitz constant of $\tilde{u}$ in $x$. Since it can be
easily derived from (2.2) that $|w(t,x_{1})-w(t,x_{2})|\leq M|x_{1}-x_{2}|$
for any given $(t,x_{1}),\ (t,x_{2})\in\Pi_{T}$, we have

$\displaystyle|\tilde{u}(t,x_{1})-\tilde{u}(t,x_{2})|$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{t}\tilde{u}_{t}(s,x_{1})-\tilde{u}_{t}(s,x_{2}){\rm
d}s+u_{0}(x_{1})-u_{0}(x_{2})\right|$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}|\tilde{u}_{t}(s,x_{1})-\tilde{u}_{t}(s,x_{2})|{\rm
d}s+L_{0}|x_{1}-x_{2}|$ $\displaystyle\leq$ $\displaystyle
L_{0}|x_{1}-x_{2}|+\int_{0}^{t}|f_{1}(\tilde{u}(s,x_{1}),w(s,x_{1}))-f_{1}(\tilde{u}(s,x_{2}),w(s,x_{2}))|{\rm
d}s$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}(a+b)|\tilde{u}(s,x_{1})-\tilde{u}(s,x_{2})|{\rm
d}s+(bA_{1}TM+L_{0})|x_{1}-x_{2}|.$

Then noticing $0<T\leq 1$ and making use of the Gronwall inequality, we obtain

$\displaystyle|\tilde{u}(t,x_{1})-\tilde{u}(t,x_{2})|\leq(bA_{1}M+L_{0})e^{a+b}|x_{1}-x_{2}|.$

This shows that $L_{\tilde{u}}=(bA_{1}M+L_{0})e^{a+b}$ is the Lipschitz
constant of $\tilde{u}$. Define

$\mathbb{Y}_{u_{0}}^{T}=\left\\{\phi\in C(\Pi_{T}):\phi(0,x)=u_{0}(x),\
0\leq\phi\leq A_{1},\ |\phi(t,x)-\phi(t,y)|\leq L_{\tilde{u}}|x-y|\right\\}.$

Obviously, $\mathbb{Y}_{u_{0}}^{T}$ is complete with the metric
$d(\phi_{1},\phi_{2})=\sup_{(t,x)\in\Pi_{T}}|\phi_{1}(t,x)-\phi_{2}(t,x)|$.
The above analysis allows us to define a map $\mathcal{F}(u)=\tilde{u}$, and
$\mathcal{F}$ maps $\mathbb{Y}_{u_{0}}^{T}$ into itself.

Step 3: We are in the position to prove that $\mathcal{F}$ is a contraction
mapping in $Y_{u_{0}}^{T}$ for sufficiently small $T$. In fact, for $i=1,2$,
let $v_{i},w_{i},g_{i},h_{i}$ be the unique solution of (2.1) with $u=u_{i}$.
By arguing as in the proof of [14, Theorem 1.1], we can show that there exists
a constant $L_{v}$, which only depends on $\mathcal{A}$, such that for any
given $(t,x_{1}),(t,x_{2})\in\overline{D}^{T}_{g_{i},h_{i}}$,

$\displaystyle|v_{i}(t,x_{1})-v_{i}(t,x_{2})|\leq L_{v}|x_{1}-x_{2}|.$ (2.4)

Denote $U=u_{1}-u_{2}$, $\tilde{U}=\tilde{u}_{1}-\tilde{u}_{2}$,
$V=v_{1}-v_{2}$ and $W=w_{1}-w_{2}$. Since $\tilde{u}_{i}$ satisfy

$\displaystyle\left\\{\begin{aligned}
&\tilde{u}_{i,t}=f_{1}(\tilde{u}_{i},w_{i}),&&(t,x)\in(0,T]\times\mathbb{R},\\\
&\tilde{u}_{i}(0,x)=u_{0}(x)>0,&&x\in\mathbb{R},\end{aligned}\right.$

it follows that, for any $(t,x)\in\Pi_{T}$,

$\displaystyle|\tilde{U}(t,x)|\leq\int_{0}^{t}(a+b)|\tilde{U}(s,x)|{\rm
d}s+bA_{1}T\|W\|_{L^{\infty}(\Pi_{T})}.$

By virtue of the Gronwall inequality again, it yields

$\displaystyle|\tilde{U}(t,x)|\leq bA_{1}e^{a+b}T\|W\|_{L^{\infty}(\Pi_{T})}.$
(2.5)

The following arguments are devoted to an estimate of
$\|W\|_{L^{\infty}(\Pi_{T})}$. Evidently, $w_{i}$ satisfy

$\displaystyle\left\\{\begin{aligned}
&w_{i,t}-dw_{i,xx}=f_{3}(v_{i},w_{i}),&&0<t\leq T,~{}g_{i}(t)<x<h_{i}(t),\\\
&w_{i}(t,x)=0,&&0<t\leq T,~{}x\notin(g_{i}(t),h_{i}(t)),\\\
&w_{i}(0,x)=w_{0}(x),&&|x|\leq h_{0}.\end{aligned}\right.$

We straighten the boundaries and define

$x_{i}(t,y)=\frac{(h_{i}(t)-g_{i}(t))y+h_{i}(t)+g_{i}(t)}{2},\ \
z_{i}(t,y)=w_{i}(t,x_{i}(t,y)),\ \ r_{i}(t,y)=v_{i}(t,x_{i}(t,y)).$

For simplicity, we introduce the following notations $\xi=\xi_{1}-\xi_{2}$,
$\zeta=\zeta_{1}-\zeta_{2}$, $z=z_{1}-z_{2}$, $r=r_{1}-r_{2}$,
$h=h_{1}-h_{2}$, $g=g_{1}-g_{2}$, where

$\displaystyle\xi_{i}(t)=\displaystyle\frac{4}{(h_{i}(t)-g_{i}(t))^{2}},\ \
\zeta_{i}(t,y)=\displaystyle\frac{h^{\prime}_{i}(t)+g^{\prime}_{i}(t)}{h_{i}(t)-g_{i}(t)}+\displaystyle\frac{(h^{\prime}_{i}(t)-g^{\prime}_{i}(t))y}{h_{i}(t)-g_{i}(t)}.$

Then $z$ satisfies

$\displaystyle\left\\{\begin{aligned}
&z_{t}-d\xi_{1}z_{yy}-\zeta_{1}z_{y}=d\xi z_{2,yy}+\zeta
z_{2,y}+\frac{kr}{1+z_{1}}-\frac{kr_{2}z}{(1+z_{1})(1+z_{2})}-qz,&&0<t\leq
T,~{}|y|<1,\\\ &z(t,\pm 1)=0,&&0\leq t\leq T,\\\ &z(0,y)=0,&&|y|\leq
1.\end{aligned}\right.$

By the $L^{p}$ estimates for parabolic equations, we see

$\|z\|_{W^{1,2}_{p}(\Delta_{T})}\leq
C_{1}\big{(}\|g,\,h\|_{C^{1}([0,T])}+\|r\|_{C(\Delta_{T})}\big{)}.$

We now estimate $\|r\|_{C(\Delta_{T})}$. For any given $(t,y)\in\Delta_{T}$,
it follows that

$\displaystyle|r(t,y)|=|v_{1}(t,x_{1}(t,y))-v_{2}(t,x_{2}(t,y))|\leq|V(t,x_{1}(t,y))|+|v_{2}(t,x_{1}(t,y))-v_{2}(t,x_{2}(t,y))|.$

It follows from the inequality (2.4) that

$|v_{2}(t,x_{1}(t,y))-v_{2}(t,x_{2}(t,y))|\leq C_{2}\|g,h\|_{C^{1}([0,T])}.$

Additionally, we can prove the following inequality:

$\displaystyle|V(t,x_{1}(t,y))|\leq
C_{3}\left(\|g,h\|_{C^{1}([0,T])}+T\|U,\,W\|_{L^{\infty}(\Pi_{T})}\right).$
(2.6)

Its proof will be put in the next step on account of the length. Thus we have

$\displaystyle\|r\|_{C(\Delta_{T})}\leq
C_{4}\left(\|g,\,h\|_{C^{1}([0,T])}+T\|U,\,W\|_{L^{\infty}(\Pi_{T})}\right).$

Then it follows that

$\|z\|_{W^{1,2}_{p}(\Delta_{T})}\leq
C_{5}\left(\|g,\,h\|_{C^{1}([0,T])}+T\|U,\,W\|_{L^{\infty}(\Pi_{T})}\right).$

By utilizing the similar methods in Step 2 of [26, Theorem 2.1] and the
embedding theorem: $[z_{y}]_{C^{\frac{\alpha}{2},\alpha}(\Delta_{T})}\leq
C\|z\|_{W^{1,2}_{p}(\Delta_{T})}$ for some positive constant $C$ independent
of $T^{-1}$ ([27, Theorem 1.1]), we can show that

$\|W\|_{L^{\infty}(\Pi_{T})}\leq
C_{6}\left(T\|W\|_{L^{\infty}(\Pi_{T})}+\|U\|_{L^{\infty}(\Pi_{T})}\right).$

Hence

$\|W\|_{L^{\infty}(\Pi_{T})}\leq 2C_{6}\|U\|_{L^{\infty}(\Pi_{T})}\ \ \
\mbox{if}\ \ 0<T\ll 1.$

This combined with (2.5) arrives at

$\|\tilde{U}\|_{L^{\infty}(\Pi_{T})}\leq
C_{7}T\|U\|_{L^{\infty}(\Pi_{T})}\leq\displaystyle\frac{1}{2}\|U\|_{L^{\infty}(\Pi_{T})}\
\ \ \mbox{if}\ \ 0<T\ll 1.$

As a consequence, $\mathcal{F}$ is a contraction mapping and there exists a
unique local solution $(u,v,w,g,h)$. Moreover the desired properties of the
local solution can be obtained from the above arguments.

Step 4: In this step, we are going to tackle the estimate (2.6), which will be
divided into several cases. By the definition of $x_{1}(t,y)$, it is easy to
see that $g_{1}(t)\leq x_{1}(t,y)\leq h_{1}(t)$. We denote $x_{1}=x_{1}(t,y)$
for simplicity.

Case 1: $x_{1}\notin(g_{2}(t),h_{2}(t))$. In this case $v_{2}(t,x_{1})=0$, and
either $g_{1}(t)\leq x_{1}\leq g_{2}(t)$ or $h_{2}(t)\leq x_{1}\leq h_{1}(t)$.
We only deal with the former case. Hence

$\displaystyle|V(t,x_{1})|$
$\displaystyle=|v_{1}(t,x_{1})-v_{1}(t,g_{1}(t))|\leq L_{v}|x_{1}-g_{1}(t)|$
$\displaystyle\leq L_{v}|g_{2}(t)-g_{1}(t)|\leq L_{v}\|g\|_{C^{1}([0,T])}.$

Case 2: $x_{1}\in(g_{2}(t),h_{2}(t))$ and either $x_{1}>h_{0}$ or
$x_{1}<-h_{0}$. We deal with only the case $x_{1}>h_{0}$. Then we can uniquely
find $0<t_{x_{1}},t^{\prime}_{x_{1}}\leq t$ such that $h_{1}(t_{x_{1}})=x_{1}$
and $h_{2}(t^{\prime}_{x_{1}})=x_{1}$. Then
$v_{1}(t_{x_{1}},x_{1})=v_{2}(t^{\prime}_{x_{1}},x_{1})=0$. Without loss of
generality, we assume $t^{\prime}_{x_{1}}>t_{x_{1}}$. Then
$h_{1}(t_{x_{1}}^{\prime})>h_{1}(t_{x_{1}})=x_{1}=h_{2}(t_{x_{1}}^{\prime})$,
$x_{1}\in(g_{1}(s),h_{1}(s))\cap(g_{2}(s),h_{2}(s))$ for all
$t_{x_{1}}^{\prime}<s\leq t$ and
$x_{1}\in(g_{1}(t_{x_{1}}^{\prime}),h_{1}(t_{x_{1}}^{\prime}))\setminus(g_{2}(t_{x_{1}}^{\prime}),h_{2}(t_{x_{1}}^{\prime}))$.
Hence,

$|V(t_{x_{1}}^{\prime},x_{1})|=v_{1}(t_{x_{1}}^{\prime},x_{1})\leq
L_{v}\|g,\,h\|_{C([0,T])}$

by the conclusion of Case 1. Integrating the differential equation of $v_{i}$
from $t_{x_{1}}^{\prime}$ to $s\,(t_{x_{1}}^{\prime}<s\leq t)$ we obtain

$\displaystyle v_{1}(s,x_{1})$ $\displaystyle=$ $\displaystyle
v_{1}(t_{x_{1}}^{\prime},x_{1})+\int_{t_{x_{1}}^{\prime}}^{s}f_{2}(u_{1},v_{1},w_{1})\big{|}_{x=x_{1}}{\rm
d}\tau,$ $\displaystyle v_{2}(s,x_{1})$ $\displaystyle=$
$\displaystyle\int_{t_{x_{1}}^{\prime}}^{s}f_{2}(u_{2},v_{2},w_{2})\big{|}_{x=x_{1}}{\rm
d}\tau.$

It then follows that

$\displaystyle|V(s,x_{1})|$ $\displaystyle\leq$ $\displaystyle
v_{1}(t_{x_{1}}^{\prime},x_{1})+\int_{t_{x_{1}}^{\prime}}^{s}\left|\frac{bu_{1}w_{1}}{1+w_{1}}-\frac{bu_{2}w_{2}}{1+w_{2}}+c(v_{2}-v_{1})\right|_{x=x_{1}}{\rm
d}\tau$ $\displaystyle\leq$ $\displaystyle
L_{v}\|g,\,h\|_{C([0,T])}+TC_{8}\big{(}\|V(\cdot,x_{1})\|_{C[t^{\prime}_{x_{1}},t])}+\|U,\,W\|_{L^{\infty}(\Pi_{T})}\big{)},$

where $C_{8}=\max\\{c,\,bA_{1},\,b\\}$. It follows from that

$|V(t,x_{1})|\leq
C_{10}\left(\|g,\,h\|_{C^{1}([0,T])}+T\|U,\,W\|_{L^{\infty}(\Pi_{T})}\right)$

if $T>0$ is sufficiently small.

Case 3: $x_{1}\in(g_{2}(t),h_{2}(t))$ and $x_{1}\in[-h_{0},h_{0}]$. In this
case we can derive

$|V(t,x_{1})|\leq C_{11}T\|U,\,W\|_{L^{\infty}(\Pi_{T})}$

by using similar methods. Since it is actually much simpler, we omit the
details. In conclusion, we have proved the estimate (2.6). ∎

######  Theorem 2.2.

(Global solution) The problem (1.2) has a unique global solution
$(u,v,w,g,h)$, and there exist four positive constants $A_{i}$, $i=1,2,3,4$,
such that

$\displaystyle(u,v,w,g,h)\in C^{1,1-}(\Pi_{\infty})\times
C^{1,1-}(\Pi_{\infty})\times
W^{1,2}_{p}(D^{\infty}_{g,h})\times[C^{1+\frac{\alpha}{2}}([0,\infty])]^{2},$
$\displaystyle 0<u\leq A_{1}\ {\rm in}\ \Pi_{\infty};\ \ 0<v\leq A_{2},\
0<w\leq A_{3}\ {\rm in}\ D^{\infty}_{g,h};\ \
0<-g^{\prime}(t),h^{\prime}(t)\leq A_{4}\ {\rm in}\ [0,\infty),$

where $A_{1}=\max\left\\{\|u_{0}\|_{\infty},\,\theta/a\right\\}$.

###### Proof.

It follows from Theorem 2.1 that the problem (1.2) has a unique local solution
$(u,v,w,g,h)$ for some $0<T\ll 1$ and $g^{\prime}(t)<0$, $h^{\prime}(t)>0$ for
$0\leq t\leq T$.

It is easy to show that $0<u\leq A_{1}$ in $\Pi_{T}$. Recalling the equations
of $(v,w)$ we can readily conclude that there exists $A_{2},\ A_{3}>0$ such
that $0<v\leq A_{2}$, $0<w\leq A_{3}$ in $D^{T}_{g,h}$. Making use of the
similar arguments in the proof of [28, Lemma 2.1], we can show that there
exists constant $A_{4}>0$ , which only depends on the initial data, such that
$0<-g^{\prime}(t),\,h^{\prime}(t)\leq A_{4}$ in $[0,T]$.

With above estimates, we can extend the local solution uniquely to the global
solution, and

$\displaystyle(u,v,w,g,h)\in C^{1,1-}(\Pi_{\infty})\times
C^{1,1-}(\overline{D}^{\infty}_{g,h})\times
W^{1,2}_{p}(D^{\infty}_{g,h})\times[C^{1+\frac{\alpha}{2}}([0,\infty))]^{2};$

see [27, Corollary 1.1] for the details. It follows from the standard
parabolic regularity theory that $(u,v,w,g,h)$ is the unique classical
solution of (1.2). Combining $v(t,x)=0$ for $x\notin(g(t),h(t))$ and the
equation satisfied by $v$, we easily derive that $v\in
C^{1,1-}(\Pi_{\infty})$. The proof is ended. ∎

Since $g^{\prime}(t)<0$, $h^{\prime}(t)>0$, there exist
$g_{\infty}\in[-\infty,0)$ and $h_{\infty}\in(0,\infty]$ such that

$\lim_{t\to\infty}g(t)=g_{\infty},\ \ \ \lim_{t\to\infty}h(t)=h_{\infty}.$

The case $h_{\infty}=-g_{\infty}=\infty$ is called Spreading, and the case
$h_{\infty}-g_{\infty}<\infty$ is called Vanishing.

######  Theorem 2.3.

(Uniform estimates) Let $(u,v,w,g,h)$ be the unique global solution of (1.2).
Then there exists a constant $C>0$ such that

$\displaystyle\|w(t,\cdot)\|_{C^{1}([g(t),h(t)])}\leq C,\ \
\|g^{\prime},\,h^{\prime}\|_{C^{{\alpha}/2}([1,\infty))}\leq C,\ \ \forall\
t\geq 1.$ (2.7)

###### Proof.

Remember $0\leq v\leq A_{2}$, $0\leq w\leq A_{3}$. The estimates (2.7) can be
proved by using analogous methods in [29, Theorem 2.1] for the case
$h_{\infty}-g_{\infty}<\infty$ and [25, Theorem 2.2] for the case
$h_{\infty}-g_{\infty}=\infty$. We omit the details here. ∎

##  3 Preliminaries

In this section, we will show some preliminaries which are crucial in the
later parts. First we will investigate an eigenvalue problem and analyze the
properties of its principal eigenvalue which will pave the ground for later
discussion. It is well known that the eigenvalue problem

$\displaystyle\left\\{\begin{aligned}
&d\hat{\phi}_{xx}+a_{11}\hat{\phi}=\rho\hat{\phi},&&l_{1}<x<l_{2},\\\
&\hat{\phi}(l_{i})=0,&&i=1,2\end{aligned}\right.$

has a principle eigenpair $(\rho_{1},\hat{\phi}_{1})$, where

$\rho_{1}=a_{11}-\displaystyle\frac{d\pi^{2}}{(l_{2}-l_{1})^{2}},~{}~{}~{}\hat{\phi}_{1}(x)=\cos\displaystyle\frac{\pi(2x-l_{2}-l_{1})}{2(l_{2}-l_{1})}.$

Now we consider the following eigenvalue problem

$\displaystyle\left\\{\begin{aligned}
&d\phi_{xx}+a_{11}\phi+a_{12}\psi=\lambda\phi,&&l_{1}<x<l_{2},\\\
&a_{21}\phi+a_{22}\psi=\lambda\psi,&&l_{1}<x<l_{2},\\\
&\phi(l_{i})=0,&&i=1,2\end{aligned}\right.$ (3.1)

with $a_{12}$, $a_{21}>0$ and $a_{11}$, $a_{22}<0$. It is clear that if
$(\lambda,(\phi,\psi))$ is an eigenpair of (3.1), then $\lambda\not=a_{22}$
and $\psi(l_{i})=0$, $i=1,2$. Define

$\mathcal{L}=\begin{pmatrix}d\partial_{xx}+a_{11}&&a_{12}\\\
a_{21}&&a_{22}\end{pmatrix},$

and choose the domain of $\mathcal{L}$:

$\mathcal{D}\big{(}\mathcal{L}\big{)}=\left\\{(\phi,\psi)\in
H^{2}((l_{1},l_{2}))\times L^{2}((l_{1},l_{2})):\phi(l_{i})=0,\
i=1,2\right\\}.$

Similarly to the proof of [15, Theorem 3.1], we can prove the following
results by means of [30, Theorem 2.3, Remark 2.2]. The details are omitted
here.

######  Theorem 3.1.

Let $\sigma(\mathcal{L})$ be the spectral set of $\mathcal{L}$ and
$\mathfrak{s}(\mathcal{L}):=\sup\\{Re\lambda:\lambda\in\sigma(\mathcal{L})\\}$.
Then the following properties hold true:

(i) $\mathfrak{s}(\mathcal{L})$ is the principal eigenvalue of (3.1) with
positive eigenvectors $(\phi_{1},\psi_{1})$;

(ii)
$\mathfrak{s}(\mathcal{L})=\displaystyle\frac{1}{2}\big{[}\rho_{1}+a_{22}+\sqrt{(\rho_{1}-a_{22})^{2}+4a_{12}a_{21}}\big{]}$
and has the same sign with $\rho_{1}-a_{12}a_{21}/{a_{22}}$;

(iii) $\mathfrak{s}(\mathcal{L})$ is strictly monotone increasing in the
length of the interval $(l_{1},l_{2})$ and strictly monotone decreasing in
$d$.

By Theorem 3.1, we can easily deduce the following result.

######  Corollary 3.2.

Define $\Gamma=a_{11}-\frac{a_{12}a_{21}}{a_{22}}$. Let $\lambda_{1}$ be the
principal eigenvalue of the problem (3.1). Then the following properties are
valid:

(i) If $\Gamma\leq 0$, then $\lambda_{1}<0$ for any $d>0$ and $(l_{1},l_{2})$;

(ii) If $\Gamma>0$, we fix the domain $(l_{1},l_{2})$ and let
$d^{*}(l_{1},l_{2})=\Gamma(l_{2}-l_{1})^{2}\pi^{-2}$. Then $\lambda_{1}>0$
when $0<d<d^{*}(l_{1},l_{2})$, $\lambda_{1}=0$ when $d=d^{*}(l_{1},l_{2})$,
and $\lambda_{1}<0$ when $d>d^{*}(l_{1},l_{2})$;

(iii) If $\Gamma>0$, we fix $d>0$ and set $L^{*}(d)=\pi\sqrt{d/\Gamma}$. Then
$\lambda_{1}>0$ when $l_{2}-l_{1}>L^{*}(d)$, $\lambda_{1}=0$ when
$l_{2}-l_{1}=L^{*}(d)$ and $\lambda_{1}<0$ when $l_{2}-l_{1}<L^{*}(d)$.

Let $\lambda_{1}$ be the principal eigenvalue of (3.1), that is, two
components of the corresponding eigenfunction are both positive or negative.
Then we have

$\lambda_{1}>a_{22},\
\rho_{1}=\lambda_{1}-{a_{12}a_{21}}/({\lambda_{1}-a_{22}}).$

Thus by the uniqueness of $\rho_{1}$ we easily derive the uniqueness of the
principal eigenvalue of (3.1).

Let $(\mu_{1},u_{1})$ be the first eigenpair of $-\Delta$ with homogeneous
Dirichlet boundary condition on $(l_{1},l_{2})$ and
($\lambda_{1},(\phi_{1},\psi_{1}))$ be the principal eigenpair of the problem
(3.1). The direct calculation yields

$\displaystyle\begin{pmatrix}-d\mu_{1}+a_{11}&&a_{12}\\\
a_{12}&&a_{22}\end{pmatrix}\begin{pmatrix}\langle\phi_{1},u_{1}\rangle\\\
\langle\psi_{1},u_{1}\rangle\end{pmatrix}=\lambda_{1}\begin{pmatrix}\langle\phi_{1},u_{1}\rangle\\\
\langle\psi_{1},u_{1}\rangle\end{pmatrix},$ (3.2)

where $\langle\cdot,\cdot\rangle$ denotes the inner product in
$L^{2}((l_{1},l_{2}))$.

The following lemma will play an important role in the study of long time
behaviors of $(u,v,w)$ when $h_{\infty}-g_{\infty}<\infty$.

######  Lemma 3.3.

Let $m(t,x)$ be a bounded function, $d$, $C$, $\mu$ and $\eta_{0}$ be positive
constants, and constant $x_{0}<\eta_{0}$. Let $\eta\in C^{1}([0,\infty))$,
$w\in W^{1,2}_{p}(D_{T})$ and $w_{0}\in W^{2}_{p}((x_{0},\eta_{0}))$ for any
$T>0$ with some $p>1$, and $w_{x}\in C(D)$, where
$D_{T}=\\{(t,x):0<t<T,x_{0}<x<\eta(t)\\}$, $D=\\{(t,x):0\leq
t\infty,x_{0}<x\leq\eta(t)\\}$. Assume that $(w,\eta)$ satisfies

$\displaystyle\left\\{\begin{array}[]{lll}w_{t}-dw_{xx}+m(t,x)w_{x}\geq-
Cw,&t>0,\ \ x_{0}<x<\eta(t),\\\\[1.42262pt] w\geq 0,&t>0,\ \
x=x_{0},\\\\[1.42262pt] w=0,\ \ \eta^{\prime}(t)\geq-\mu w_{x},&t>0,\ \
x=\eta(t),\\\\[1.42262pt] w(0,x)=w_{0}(x)\geq,\,\not\equiv
0,&x\in(x_{0},\eta_{0}),\\\\[1.42262pt] \eta(0)=\eta_{0},\end{array}\right.$

and $\displaystyle\lim_{t\to\infty}\eta(t)=\eta_{\infty}<\infty$,
$\displaystyle\lim_{t\to\infty}\eta^{\prime}(t)=0$,

$\|w(t,\cdot)\|_{C^{1}([x_{0},\,\eta(t)])}\leq M,\ \,\forall\,t\geq 1$

for some constant $M>0$. Then $\displaystyle\lim_{t\to\infty}\,\max_{x_{0}\leq
x\leq\eta(t)}w(t,x)=0$.

###### Proof.

When $x_{0}=0$ and $m(t,x)=0$, this lemma is exactly Proposition 2 in [31];
when $x_{0}=0$ and $m(t,x)=\gamma$ is a constant, this lemma is exactly Lemma
3.1 in [32]. For our present case, by the maximum principle we have $w(t,x)>0$
for $t>0$ and $x_{0}<x<\eta(t)$. If we follow the proof of [21, Theorem 2.2]
word by word we can prove this lemma. We will leave out the details because
the advection term and boundary condition at $x=x_{0}$ do not influence the
availability of the argument in [21, Theorem 2.2]. ∎

######  Lemma 3.4.

$($Comparison principle$)$ Let $T>0$, $\bar{g},\bar{h}\in C^{1}([0,T])$ and
$\bar{g}<\bar{h}$ in $[0,T]$. Let $\bar{u}\in C^{1,0}([0,T]\times\mathbb{R})$,
$\bar{v}\in C^{1,0}(\overline{O}),\bar{w}\in C(\overline{O})\cap C^{1,2}(O)$
with $O=\\{0<t\leq T,\,\bar{g}(t)<x<\bar{h}(t)\\}$. Assume that
$(\bar{u},\bar{v},\bar{w},\bar{g},\bar{h})$ satisfies

$\displaystyle\left\\{\begin{aligned}
&\bar{u}_{t}\geq\theta-a\bar{u},&&t>0,~{}-\infty<x<\infty,\\\ &\bar{v}_{t}\geq
f_{2}(\bar{u},\bar{v},\bar{w}),&&t>0,~{}\bar{g}(t)<x<\bar{h}(t),\\\
&\bar{w}_{t}-d\bar{w}_{xx}\geq
f_{3}(\bar{v},\bar{w}),&&t>0,~{}\bar{g}(t)<x<\bar{h}(t),\\\
&\bar{v}(t,x)=\bar{w}(t,x)=0,&&t>0,\ x=\bar{g}(t),\,\bar{h}(t),\\\
&\bar{g}^{\prime}(t)\leq-\mu\bar{w}_{x}(t,\bar{g}(t)),&&t\geq 0,\\\
&\bar{h}^{\prime}(t)\geq-\beta\bar{w}_{x}(t,\bar{h}(t)),&&t\geq 0,\\\
&\bar{v}(0,x),\bar{w}(0,x)\geq 0,&&\bar{g}(0)\leq
x\leq\bar{h}(0).\end{aligned}\right.$

If $\bar{g}(0)\leq-h_{0},\,\bar{h}(0)\geq h_{0}$, $u_{0}(x)\leq\bar{u}(0,x)$
in $\mathbb{R}$, and $v_{0}(x)\leq\bar{v}(0,x),w_{0}(x)\leq\bar{w}(0,x)$ on
$[-h_{0},h_{0}]$. Then the solution $(u,v,w,g,h)$ of (1.2) satisfies

$\displaystyle g\geq\bar{g},\ h\leq\bar{h}\ \ {\rm on}\ \,[0,T];\ \
u\leq\bar{u}\ \ {\rm on}\ \,[0,T]\times\mathbb{R};\ \ v\leq\bar{v},\
w\leq\bar{w}\ \ {\rm on}\ \,\overline{D}_{g,h}^{T},$

where $D_{g,h}^{T}$ is defined as in the beginning of Section 2.

###### Proof.

Take $0<\rho<1$ and let $(u_{\rho},v_{\rho},w_{\rho},g_{\rho},h_{\rho})$ be
the corresponding unique solution of (1.2) with $(h_{0},v_{0},w_{0})$ replaced
by $(\rho h_{0},v_{0,\rho},w_{0,\rho})$, where $v_{0,\rho}(x),w_{0,\rho}(x)$
satisfy (1.3) with $h_{0}$ replaced by $\rho h_{0}$, and satisfy

$\displaystyle 0<v_{0,\rho}(x)\leq v_{0}(x),\ 0<w_{0,\rho}(x)\leq w_{0}(x)\ \
\,{\rm on}\ \ (-\rho h_{0},\rho h_{0}),$

as well as

$\displaystyle\lim_{\rho\to 1}v_{0,\rho}\left(\rho x\right)=v_{0}(x)\ \ {\rm
in}\ \ W_{\infty}^{1}((-h_{0},h_{0})),\ \ \ \lim_{\rho\to
1}w_{0,\rho}\left(\rho x\right)=w_{0}(x)\ \ \,{\rm in}\ \
W_{p}^{2}((-h_{0},h_{0})).$

By a simple comparison consideration, we have $u_{\rho}\leq\bar{u}$ on
$[0,T]\times\mathbb{R}$. Thus $(v_{\rho},w_{\rho})$ satisfies

$\displaystyle\left\\{\begin{aligned} &v_{\rho,t}\leq
f_{2}(\bar{u},v_{\rho},w_{\rho}),&&t>0,~{}g_{\rho}(t)<x<h_{\rho}(t),\\\
&w_{\rho,t}-dw_{\rho,xx}=f_{3}(v_{\rho},w_{\rho}),&&t>0,~{}g_{\rho}(t)<x<h_{\rho}(t),\\\
&v_{\rho}(t,x)=w_{\rho}(t,x)=0,&&t>0,\ x\notin(g_{\rho}(t),h_{\rho}(t)),\\\
&g^{\prime}_{\rho}(t)=-\mu
w_{\rho,x}(t,g_{\rho}(t)),\,\,\,h^{\prime}_{\rho}(t)=-\beta
w_{\rho,x}(t,h_{\rho}(t)),&&t\geq 0,\\\\[1.42262pt]
&v_{\rho}(0,x)=v_{0,\rho}(x),\ \ \ w_{\rho}(0,x)=w_{0,\rho}(x),&&-\rho
h_{0}\leq x\leq\rho h_{0},\\\\[1.42262pt] &h_{\rho}(0)=-g_{\rho}(0)=\rho
h_{0}.\end{aligned}\right.$

Similarly to [17, Lemma 3.5], by use of the indirect arguments and strong
maximum principle we can show that
$g_{\rho}(t)>\bar{g}(t),h_{\rho}(t)<\bar{h}(t)$ for $0\leq t\leq T$. Thus
$v_{\rho}(t,x)<\bar{v}(t,x),w_{\rho}(t,x)<\bar{w}(t,x)$ for $0<t\leq T$ and
$g_{\rho}(t)\leq x\leq h_{\rho}(t)$ by the standard comparison principle.
Letting $\rho\to 1$ and using the continuous dependence of solutions on
parameters we have
$(u_{\rho},v_{\rho},w_{\rho},g_{\rho},h_{\rho})\to(u,v,w,g,h)$. The details
are omitted. ∎

##  4 Long time behavior of $(u,v,w)$

This section concerns with the long time behavior of $(u,v,w)$. We first study
the vanishing case ($h_{\infty}-g_{\infty}<\infty$).

######  Theorem 4.1.

Let $(u,v,w,g,h)$ be the unique global solution of (1.2). If
$h_{\infty}-g_{\infty}<\infty$, then

$\displaystyle\displaystyle\lim_{t\to\infty}\|w(t,\cdot)\|_{C([g(t),\,h(t)])}=\lim_{t\to\infty}\|v(t,\cdot)\|_{C([g(t),\,h(t)])}=0,$
$\displaystyle\displaystyle\lim_{t\to\infty}u(t,x)=\theta/a\ \ \mbox{
uniformly\, in }\ \mathbb{R}.$

###### Proof.

By the second estimate in (2.7), we see that both $g^{\prime}(t)$ and
$h^{\prime}(t)$ are uniformly continuous in $[1,\infty)$. Since
$h_{\infty}-g_{\infty}<\infty$, it is easy to deduce
$\displaystyle\lim_{t\rightarrow\infty}g^{\prime}(t)=\lim_{t\rightarrow\infty}h^{\prime}(t)=0$.
Then, using the first estimate of (2.7), Lemma 3.3 in $[0,h(t))$ and a similar
version of Lemma 3.3 in $(g(t),0]$, one can arrive at
$\lim\limits_{t\to\infty}\|w(t,\cdot)\|_{C([g(t),\,h(t)])}=0$. For any
$\varepsilon>0$, there exists $T>0$ such that
$bA_{1}w(t,x)/(1+w(t,x))\leq\varepsilon$ for $t\geq T$ and $x\in\mathbb{R}$.
Thus $v$ satisfies

$\displaystyle\left\\{\begin{aligned} &v_{t}\leq\varepsilon-cv,&&t\geq
T,~{}g(t)<x<h(t),\\\ &v(t,g(t))=v(t,h(t))=0,&&t\geq T,\\\ &v(T,x)\geq
0.\end{aligned}\right.$

By the comparison principle, we have
$\limsup\limits_{t\to\infty}\|v(t,\cdot)\|_{C([g(t),\,h(t)])}\leq{\varepsilon}/{c}$.
The arbitrariness of $\varepsilon$ implies
$\lim\limits_{t\to\infty}\|v(t,\cdot)\|_{C([g(t),\,h(t)])}=0$. Similarly, we
can easily deduce that

$\displaystyle\displaystyle\limsup_{t\to\infty}u(t,x)\leq{\theta}/{a}\ \ {\rm
uniformly~{}in~{}}\mathbb{R}.$ (4.1)

On the other hand, for any $\varepsilon_{1}>0$, there exists $T_{1}>0$ such
that $w(t,x)/(1+w(t,x))\leq\varepsilon_{1}$ for $t\geq T_{1}$ and
$x\in\mathbb{R}$. So $u$ satisfies

$\displaystyle\left\\{\begin{aligned}
&u_{t}\geq\theta-(a+b\varepsilon_{1})u,&&t\geq T_{1},~{}x\in\mathbb{R},\\\
&u(T_{1},x)>0,&&x\in\mathbb{R}.\end{aligned}\right.$

Let $\underline{u}$ be the unique solution of the problem

$\displaystyle\left\\{\begin{aligned}
&\underline{u}_{t}=\theta-(a+b\varepsilon_{1})\underline{u},&&t\geq T_{1},\\\
&\underline{u}(T_{1})=0.\end{aligned}\right.$

By using comparison principle and the fact that
$\displaystyle\lim_{t\to\infty}\underline{u}(t)=\theta/(a+b\varepsilon_{1})$,
we have that
$\displaystyle\liminf_{t\to\infty}u(t,\cdot)\geq\theta/(a+b\varepsilon_{1})$
uniformly in $\mathbb{R}$. Due to the arbitrariness of $\varepsilon_{1}$ and
(4.1), we derive the desired result. ∎

In the following we study the spreading case ($h_{\infty}-g_{\infty}=\infty$).
To get the accurate limits of the solution components $u,v,w$ of (1.2), we
first give a proposition which concerns the existence, uniqueness and
asymptotic behavior of positive solution of a boundary value problem.

######  Proposition 4.2.

Let $m,l$ be positive constants and consider the following problem

$\displaystyle\left\\{\begin{aligned} &f_{2}(m,v,w)=0,&&-l<x<l,\\\
&-dw_{xx}=f_{3}(v,w),&&-l<x<l,\\\ &w(x)=0,&&x=\pm l.\end{aligned}\right.$
(4.2)

(i) Let $\lambda_{1}$ be the principal eigenvalue of

$\displaystyle\left\\{\begin{aligned}
&d\phi_{xx}-q\phi+k\psi=\lambda\phi,&&-l<x<l,\\\
&bm\phi-c\psi=\lambda\psi,&&-l<x<l,\\\ &\phi(\pm l)=0.\end{aligned}\right.$
(4.3)

Then (4.2) has a positive solution if and only if $\lambda_{1}>0$. Moreover,
the positive solution of (4.2) is unique when it exists.

(ii) To stress the dependence on $l$, we denote the unique positive solution
of (4.2) by $(v_{l},w_{l})$. Then $(v_{l},w_{l})$ is nondecreasing in $l$, and
converges to $(\hat{v},\hat{w})$ locally uniformly in $\mathbb{R}$ as
$l\to\infty$, where $(\hat{v},\hat{w})$ is the unique positive root of

$\displaystyle f_{2}(m,v,w)=0,\ \ \ f_{3}(v,w)=0.$ (4.4)

###### Proof.

(i) Clearly, the problem (4.2) is equivalent to

$\displaystyle\left\\{\begin{aligned} &-dw_{xx}+qw=\frac{kbmw}{c(1+w)^{2}},\ \
v=\frac{bmw}{c(1+w)},\ \ &&-l<x<l,\\\ &w(x)=0,&&x=\pm l.\end{aligned}\right.$

For clarity of exposition, we will always use the problem (4.2) in later
discussion.

If (4.2) has a positive solution $(v,w)$, it is easy to show that
$q<\lambda_{1}(q)<kbm/c$, where $\lambda_{1}(q)$ is the principal eigenvalue
of

$\displaystyle\left\\{\begin{aligned}
&-d\phi_{xx}+q\phi=\lambda\phi,&&-l<x<l,\\\ &\phi(x)=0,&&x=\pm
l.\end{aligned}\right.$ (4.5)

Moreover, since $\lambda_{1}(q)<{kbm}/{c}$, and the function ${bmk}/(c+x)-x$
is decreasing in $x>-c$, we can show that there exists the unique
$\lambda^{*}>0$ such that ${bmk}/(c+\lambda^{*})-\lambda^{*}=\lambda_{1}(q)$.
Substituting this into (4.5), one can easily see that $\lambda^{*}$ is the
principal eigenvalue of (4.3), that is, $\lambda^{*}=\lambda_{1}>0$.

If $\lambda_{1}>0$, by the standard upper and lower solution methods we can
show that (4.2) has at least one positive solution. Thanks to the structure of
nonlinear terms of (4.2), the uniqueness is easily derived.

(ii) It follows from the above analysis and Corollary 3.2 that for large $l$,
(4.2) has a unique positive solution $(v_{l},w_{l})$ provided that $kbm>qc$. A
comparison argument (Lemma 2.1 in [16]) shows that $(v_{l},w_{l})$ is
nondecreasing in $l$, and there exists $C>0$ such that $v_{l},w_{l}<C$ for all
large $l$. Making use of the standard elliptic regularity theory, we have that
$(v_{l},w_{l})\to(\tilde{v},\tilde{w})$ in $C^{2}_{loc}(\mathbb{R})$, where
$(\tilde{v},\tilde{w})$ is a positive solution of

$\displaystyle\left\\{\begin{aligned} &f_{2}(m,v,w)=0,&&-\infty<x<\infty,\\\
&-dw_{xx}=f_{3}(v,w),&&-\infty<x<\infty.\end{aligned}\right.$

Obviously, $\tilde{w}$ satisfies

$\displaystyle-d\tilde{w}_{xx}=\frac{kbm}{c(1+\tilde{w})^{2}}\tilde{w}-q\tilde{w},\
\ -\infty<x<\infty.$ (4.6)

Since $\frac{kbm}{c(1+w)^{2}}-q$ is decreasing in $w>0$, the possible positive
solution of (4.6) is a unique positive root of $kbm=qc(1+w)^{2}$. Thus
$\tilde{w}=\hat{w}$, and consequently $\tilde{v}=\hat{v}$. The proof is
finished. ∎

######  Theorem 4.3.

Suppose that $h_{\infty}=-g_{\infty}=\infty$. If
$\mathcal{R}_{0}+\sqrt{\mathcal{R}_{0}}>b/a$, then there are six positive
constants $\underline{u}_{\infty}$, $\bar{u}_{\infty}$,
$\underline{v}_{\infty}$, $\bar{v}_{\infty}$, $\underline{w}_{\infty}$ and
$\bar{w}_{\infty}$ such that

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\underline{u}_{\infty}\leq\liminf_{t\to\infty}u(t,x)\leq\limsup_{t\to\infty}u(t,x)\leq\bar{u}_{\infty},\\\
\displaystyle\underline{v}_{\infty}\leq\liminf_{t\to\infty}v(t,x)\leq\limsup_{t\to\infty}v(t,x)\leq\bar{v}_{\infty},\\\
\displaystyle\underline{w}_{\infty}\leq\liminf_{t\to\infty}w(t,x)\leq\limsup_{t\to\infty}w(t,x)\leq\bar{w}_{\infty}\end{array}\right.$
(4.10)

locally uniformly in $\mathbb{R}$. Particularly, if we assume $b\leq 2a$, then

$\displaystyle\lim_{t\to\infty}u(t,x)=u^{*},\ \
\lim_{t\to\infty}v(t,x)=v^{*},\ \ \lim_{t\to\infty}w(t,x)=w^{*}\ \ {\rm
locally~{}uniformly~{}in~{}}\mathbb{R},$ (4.11)

where $(u^{*},v^{*},w^{*})$ is a unique positive root of

$\displaystyle f_{1}(u,w)=0,\ \ f_{2}(u,v,w)=0,\ \ f_{3}(v,w)=0.$ (4.12)

###### Proof.

The condition $h_{\infty}-g_{\infty}=\infty$ implies $\mathcal{R}_{0}>1$ (cf.
Theorem 5.1). One can easily see that (4.12) has a unique positive root
$(u^{*},v^{*},w^{*})$. The following proof is actually an iterative process,
and the idea comes from [28, 33].

Step 1: Clearly,

$\limsup_{t\to\infty}u(t,x)\leq{\theta}/{a}=:\bar{u}_{1}\ \mbox{ uniformly\,
in }\ \mathbb{R}.$

Then for any $\varepsilon>0$, there exists $T>0$ such that
$u(t,x)\leq{\theta}/{a}+\varepsilon$ with $t\geq T$ and $x\in\mathbb{R}$. Thus
$(v,w)$ satisfies

$\displaystyle\left\\{\begin{aligned} &v_{t}\leq
f_{2}(\theta/a+\varepsilon,\,v,w),&&t>T,\ \ g(t)<x<h(t),\\\
&w_{t}-dw_{xx}=f_{3}(v,w),&&t>T,\ \ g(t)<x<h(t),\\\ &v(t,x)=0,\
w(t,x)=0,&&t>T,\ x=g(t)\ {\rm or}\ h(t),\\\ &v(T,x)\geq 0,\ w(T,x)\geq
0,&&g(T)\leq x\leq h(T).\end{aligned}\right.$

Consider the ODEs problem

$\displaystyle\left\\{\begin{aligned}
&\bar{v}_{t}=f_{2}(\bar{u}_{1}+\varepsilon,\,\bar{v},\bar{w}),\ \
\bar{w}_{t}=f_{3}(\bar{v},\bar{w}),&&t>T,\\\ &\bar{v}(T)=A_{2},\
\bar{w}(T)=A_{3}.\end{aligned}\right.$ (4.13)

Since ${\cal R}_{0}>1$, the problem (4.13) has a unique positive equilibrium
$(\bar{v}_{1}^{\varepsilon},\bar{w}_{1}^{\varepsilon})$ which is globally
asymptotically stable. By a simple comparison consideration, we have
$v(t,x)\leq\bar{v}(t,x)$ and $w(t,x)\leq\bar{w}(t,x)$ for $t\geq T$ and
$x\in\mathbb{R}$. And so,
$\displaystyle\limsup_{t\to\infty}v(t,x)\leq\bar{v}_{1}^{\varepsilon}$ and
$\displaystyle\limsup_{t\to\infty}w(t,x)\leq\bar{w}_{1}^{\varepsilon}$
uniformly in $\mathbb{R}$. By the arbitrariness of $\varepsilon$, we have

$\limsup_{t\to\infty}v(t,x)\leq\bar{v}_{1},\ \
\limsup_{t\to\infty}w(t,x)\leq\bar{w}_{1}\ \mbox{ uniformly\, in }\
\mathbb{R},$

where $(\bar{v}_{1},\bar{w}_{1})$ is a unique positive root of the algebraic
system (4.4) with $m$ replaced by $\bar{u}_{1}$.

Step 2: For small $\varepsilon>0$, there exists $T>0$ such that
$w(t,x)\leq\bar{w}_{1}+\varepsilon$ for $t\geq T$ and $x\in\mathbb{R}$. Hence
$u$ satisfies

$\displaystyle\left\\{\begin{aligned} &u_{t}\geq
f_{1}(u,\bar{w}_{1}+\varepsilon),&&t>T,\ x\in\mathbb{R},\\\
&u(T,x)>0,&&x\in\mathbb{R}.\end{aligned}\right.$

Using the comparison argument with the solution having initial value $0$ we
can deduce that

$\liminf_{t\to\infty}u(t,x)\geq\frac{\theta(1+\bar{w}_{1})}{a+a\bar{w}_{1}+b\bar{w}_{1}}=:\underline{u}_{1}\
\ {\rm uniformly~{}in}\ \ \mathbb{R}.$

Direct calculation shows that $\bar{w}_{1}=\sqrt{\mathcal{R}_{0}}-1$ and

$kb\underline{u}_{1}>qc\
{\rm~{}if~{}and~{}only~{}if~{}}\,\mathcal{R}_{0}+\sqrt{\mathcal{R}_{0}}>b/a.$

By our assumptions, we have $kb\underline{u}_{1}>qc$, and then
$kb(\underline{u}_{1}-\varepsilon)>qc$ for small $\varepsilon>0$.

Recall Proposition 4.2. For any large $l$, let $(v_{l},w_{l})$ and
$(\underline{v}_{1}^{\varepsilon},\underline{w}_{1}^{\varepsilon})$ be the
unique positive solutions of (4.2) and (4.4) with $m$ replaced by
$\underline{u}_{1}-\varepsilon$, respectively. Then
$(v_{l}(x),w_{l}(x))\to(\underline{v}_{1}^{\varepsilon}(x),\underline{w}_{1}^{\varepsilon}(x))$
locally uniformly in $\mathbb{R}$ as $l\to\infty$. For any given $N\gg 1$ and
$0<\sigma\ll 1$, there exists a large $l>N$ such that
$v_{l}(x)>\underline{v}_{1}^{\varepsilon}-\sigma/2$ and
$w_{l}(x)>\underline{w}_{1}^{\varepsilon}-\sigma/2$ for $x\in[-N,N]$.

For such a fixed $l>N$, let $(\lambda_{1},(\phi,\psi))$ be the principal
eigenpair of (4.3) with $m$ replaced by $\underline{u}_{1}-\varepsilon$. We
can verify that for small $\delta>0$, $(\delta\psi,\delta\phi)$ is a lower
solution of (4.2) with $m$ replaced by $\underline{u}_{1}-\varepsilon$ (see
the proof of Theorem 5.2 for details). Moreover, we may choose $T\gg 1$,
$0<\delta\ll 1$ such that $[-l,l]\subseteq(g(t),h(t))$ for $t\geq T$,
$u(t,x)\geq\underline{u}_{1}-\varepsilon$ on $[T,\infty)\times[-l,l]$, and
$\delta\psi(\cdot)\leq v(T,\cdot),\delta\phi(\cdot)\leq w(T,\cdot)$ on
$[-l,l]$. Hence $(v,w)$ satisfies

$\displaystyle\left\\{\begin{aligned} &v_{t}\geq
f_{2}(\underline{u}_{1}-\varepsilon,\,v,w),&&t>T,\ \ -l<x<l,\\\
&w_{t}-dw_{xx}=f_{3}(v,w),&&t>T,\ \ -l<x<l,\\\ &w(t,x)>0,&&t>T,\ x=\pm l,\\\
&v(T,x)\geq\delta\psi(x),\ w(T,x)\geq\delta\phi(x),&&-l\leq x\leq
l.\end{aligned}\right.$

Let $(\tilde{v},\tilde{w})$ be a unique positive solution of the following
problem

$\displaystyle\left\\{\begin{aligned}
&\tilde{v}_{t}=f_{2}(\underline{u}_{1}-\varepsilon,\,\tilde{v},\tilde{w}),&&t>T,\
\ -l<x<l,\\\ &\tilde{w}_{t}-d\tilde{w}_{xx}=f_{3}(\tilde{v},\tilde{w}),&&t>T,\
\ -l<x<l,\\\ &\tilde{w}(t,x)=0,&&t>T,\ x=\pm l,\\\
&\tilde{v}(T,x)=\delta\psi(x),\ \tilde{w}(T,x)=\delta\phi(x),&&-l\leq x\leq
l.\end{aligned}\right.$

Then $\tilde{v}$ and $\tilde{w}$ are nondecreasing in $t$. By the standard
parabolic regularity we have
$\displaystyle\lim_{t\to\infty}(\tilde{v}(t,x),\tilde{w}(t,x))=(v_{l}(x),w_{l}(x))$
uniformly in $[-l,l]$. There exists $T_{1}>T$ such that $\tilde{v}(t,x)\geq
v_{l}(x)-\sigma/2,\ \tilde{w}(t,x)\geq w_{l}(x)-\sigma/2$ for $t>T_{1}$ and
$x\in[-l,l]$. Furthermore, by the comparison principle,
$v(t,x)\geq\tilde{v}(t,x),w(t,x)\geq\tilde{w}(t,x)$ for $t>T$ and
$x\in[-l,l]$. So we have

$v(t,x)\geq\underline{v}_{1}^{\varepsilon}-\sigma,\ \
w(t,x)\geq\underline{w}_{1}^{\varepsilon}-\sigma\ \ \mbox{for}\ t>T_{1},\
|x|\leq N.$

These estimates combined with the arbitrariness of $\varepsilon$, $\sigma$ and
$N$ yield

$\liminf_{t\to\infty}v(t,x)\geq\underline{v}_{1},\ \
\liminf_{t\to\infty}w(t,x)\geq\underline{w}_{1}\ \ {\rm
locally~{}uniformly~{}in}\ \ \mathbb{R},$

where $(\underline{v}_{1},\underline{w}_{1})$ is a unique positive root of
(4.4) with $m$ replaced by $\underline{u}_{1}$.

Step 3: For any given $N>0$ and $0<\varepsilon\ll 1$, there exists $T>0$ such
that $w(t,x)\geq\underline{w}_{1}-\varepsilon$ for $t>T$ and $-N\leq x\leq N$.
So we have

$\displaystyle\left\\{\begin{aligned} &u_{t}\leq
f_{1}(u,\,\underline{w}_{1}-\varepsilon),&&t>T,\ -N\leq x\leq N,\\\
&u(T,x)>0,&&-N\leq x\leq N.\end{aligned}\right.$

Comparing with the following ODE problem

$\displaystyle\bar{u}_{t}=f_{1}(\bar{u},\,\underline{w}_{1}-\varepsilon),\ \
t>T;\ \ \ \bar{u}(T)=A_{1},$

we can show that $u(t,x)\leq\bar{u}(t)$ for $t\geq T$ and $-N\leq x\leq N$.
Similarly to the preceding arguments, we have

$\limsup_{t\to\infty}u(t,x)\leq\frac{\theta(1+\underline{w}_{1})}{a+a\underline{w}_{1}+b\underline{w}_{1}}=:\bar{u}_{2}\
\ {\rm locally~{}uniformly~{}in}\ \ \mathbb{R},$

and $\bar{u}_{2}>\underline{u}_{1}$. Moreover, the direct calculation yields
$kb\bar{u}_{2}>qc$.

For any fixed $0<\varepsilon\ll 1$, we take $K>\max\big{\\{}A_{3},\
\frac{kb(\bar{u}_{2}+\varepsilon)}{qc}\big{\\}}$ and consider the problem

$\displaystyle\left\\{\begin{aligned}
&-dw_{xx}=\frac{kb(\bar{u}_{2}+\varepsilon)w}{c(1+w)^{2}}-qw,&&-l<x<l,\\\
&w(\pm l)=K.\end{aligned}\right.$ (4.14)

Clearly, $kb(\bar{u}_{2}+\varepsilon)>qc$. By the standard method we can show
that (4.14) has a unique positive solution $w^{l}$ for large $l$. Moreover,
$0<w^{l}\leq K$. The comparison principle gives that $w^{l}$ is nonincreasing
in $l$ and $w^{l}\geq w_{l}$. In the same way as the proof of Proposition 4.2
(ii) we can derive
$\displaystyle\lim_{l\to\infty}w^{l}(x)=\bar{w}_{2}^{\varepsilon}$ locally
uniformly in $\mathbb{R}$, where $\bar{w}_{2}^{\varepsilon}$ is a unique
positive root of $kb(\bar{u}_{2}+\varepsilon)=qc(1+w)^{2}$. Take

$v^{l}(x)=\frac{b(\bar{u}_{2}+\varepsilon)w^{l}(x)}{c(1+w^{l}(x))},\ \ \
\bar{v}_{2}^{\varepsilon}=\frac{b(\bar{u}_{2}+\varepsilon)\bar{w}_{2}^{\varepsilon}}{c(1+\bar{w}_{2}^{\varepsilon})}.$

Then $\displaystyle\lim_{l\to\infty}v^{l}(x)=\bar{v}_{2}^{\varepsilon}$
locally uniformly in $\mathbb{R}$, and
$(\bar{v}_{2}^{\varepsilon},\bar{w}_{2}^{\varepsilon})$ is a unique positive
root of (4.4) in there $m$ is replaced by $\bar{u}_{2}+\varepsilon$.

For any given $N\gg 1$ and $0<\sigma\ll 1$, there exists a large $l>N$ such
that $v^{l}(x)\leq\bar{v}_{2}^{\varepsilon}+\sigma$ and
$w^{l}(x)\leq\bar{w}_{2}^{\varepsilon}+\sigma$ for $-N\leq x\leq N$. Moreover,
there exists $T>0$ such that $u(t,x)\leq\bar{u}_{2}+\varepsilon$ for
$(t,x)\in[T,\infty)\times[-l,l]$, and $h(T)>l,\ g(T)<-l$. Thanks to the
equation of $v$ and $K>A_{3}$, we can find $T_{1}>T$ such that
$v(t,x)\leq\frac{b(\bar{u}_{2}+\varepsilon)K}{c(1+K)}:=A_{2}^{*}$ on
$[T_{1},\infty)\times[-l,l]$. Therefore, $(v,w)$ satisfies

$\displaystyle\left\\{\begin{aligned} &v_{t}\leq
f_{2}(\bar{u}_{2}+\varepsilon,\,v,w),&&t>T_{1},\ \ -l<x<l,\\\
&w_{t}-dw_{xx}=f_{3}(v,w),&&t>T_{1},\ \ -l<x<l,\\\ &w(t,x)\leq K,&&t>T_{1},\
x=\pm l,\\\ &v(T_{1},x)\leq A_{2}^{*},\ w(T_{1},x)\leq K,&&-l\leq x\leq
l.\end{aligned}\right.$

Let $(\bar{v},\bar{w})$ be a unique positive solution of the problem

$\displaystyle\left\\{\begin{aligned}
&\bar{v}_{t}=f_{2}(\bar{u}_{2}+\varepsilon,\,\bar{v},\bar{w}),&&t>T_{1},\ \
-l<x<l,\\\ &\bar{w}_{t}-d\bar{w}_{xx}=f_{3}(\bar{v},\bar{w}),&&t>T_{1},\ \
-l<x<l,\\\ &\bar{w}(t,x)=K,&&t>T_{1},\ x=\pm l,\\\
&\bar{v}(T_{1},x)=A_{2}^{*},\ \bar{w}(T_{1},x)=K,&&-l\leq x\leq
l.\end{aligned}\right.$

Then we can deduce that $(\bar{v}(t,x),\bar{w}(t,x))\to(v^{l}(x),w^{l}(x))$
uniformly in $[-l,l]$ as $t\to\infty$. Thus there exists $T_{2}>T_{1}$ such
that $\bar{v}(t,x)\leq v^{l}(x)+\sigma,\bar{w}(t,x)\leq w^{l}(x)+\sigma$ for
$t>T_{2}$ and $x\in[-l,l]$. A comparison consideration yields that
$v(t,x)\leq\bar{v}(t,x)$ and $w(t,x)\leq\bar{w}(t,x)$ on
$[T_{1},\infty]\times[-l,l]$. Recalling our previous conclusion we immediately
derive that

$v(t,x)\leq\bar{v}_{2}^{\varepsilon}+2\sigma,\ \
w(t,x)\leq\bar{w}_{2}^{\varepsilon}+2\sigma,\ \ t>T_{2},-N\leq x\leq N.$

The arbitrariness of $\varepsilon$, $\sigma$ and $N$ implies

$\limsup_{t\to\infty}v(t,x)\leq\bar{v}_{2},\ \
\limsup_{t\to\infty}w(t,x)\leq\bar{w}_{2}\ \ \mbox{locally\, uniformly\, in }\
\mathbb{R},$

where $(\bar{v}_{2},\bar{w}_{2})$ is a unique positive root of the equations
(4.4) with $m$ replaced by $\bar{u}_{2}$.

We may argue as in Step 2 to conclude that

$\displaystyle\liminf_{t\to\infty}u(t,x)\geq\underline{u}_{2},\ \
\liminf_{t\to\infty}v(t,x)\geq\underline{v}_{2},\
\liminf_{t\to\infty}w(t,x)\geq\underline{w}_{2}\ \ {\rm
locally~{}uniformly~{}in}\ \ \mathbb{R},$

where
$\underline{u}_{2}=\frac{\theta(1+\bar{w}_{2})}{a+a\bar{w}_{2}+b\bar{w}_{2}}$,
and $(\underline{v}_{2},\underline{w}_{2})$ is a unique positive root of (4.4)
with $m$ replaced by $\underline{u}_{2}$.

Step 4: According to the above arguments we have

$\underline{u}_{1}<\underline{u}_{2}<\bar{u}_{2}<\bar{u}_{1},\ \
\underline{v}_{1}<\underline{v}_{2}<\bar{v}_{2}<\bar{v}_{1},\ \
\underline{w}_{1}<\underline{w}_{2}<\bar{w}_{2}<\bar{w}_{1}.$

Repeating the above procedures we can find six sequences
$\\{\underline{u}_{n}\\}$, $\\{\bar{u}_{n}\\}$, $\\{\underline{v}_{n}\\}$,
$\\{\bar{v}_{n}\\}$, $\\{\underline{w}_{n}\\}$ and $\\{\bar{w}_{n}\\}$
satisfying

$\displaystyle\underline{u}_{1}<\underline{u}_{2}<\cdots<\underline{u}_{n}<\cdots<\bar{u}_{n}<\cdots<\bar{u}_{2}<\bar{u}_{1},$
$\displaystyle\underline{v}_{1}<\underline{v}_{2}<\cdots<\underline{v}_{n}<\cdots<\bar{v}_{n}<\cdots<\bar{v}_{2}<\bar{v}_{1},$
$\displaystyle\underline{w}_{1}<\underline{w}_{2}<\cdots<\underline{w}_{n}<\cdots<\bar{w}_{n}<\cdots<\bar{w}_{2}<\bar{w}_{1},$

so that

$\displaystyle\displaystyle\underline{u}_{n}\leq\liminf_{t\to\infty}u(t,x)\leq\limsup_{t\to\infty}u(t,x)\leq\bar{u}_{n},$
$\displaystyle\displaystyle\underline{v}_{n}\leq\liminf_{t\to\infty}v(t,x)\leq\limsup_{t\to\infty}v(t,x)\leq\bar{v}_{n},$
$\displaystyle\displaystyle\underline{w}_{n}\leq\liminf_{t\to\infty}w(t,x)\leq\limsup_{t\to\infty}w(t,x)\leq\bar{w}_{n}$

locally uniformly in $\mathbb{R}$. The limits of the above six sequences are
well defined, and denoted by $\underline{u}_{\infty}$, $\bar{u}_{\infty}$,
$\underline{v}_{\infty}$, $\bar{v}_{\infty}$, $\underline{w}_{\infty}$ and
$\bar{w}_{\infty}$ respectively. It is clear that (4.10) holds.

Now we assume $b\leq 2a$ and prove (4.11). By the careful calculations one can
obtain

$\displaystyle\bar{u}_{1}=\displaystyle\frac{\theta}{a},\ \
\frac{b\bar{u}_{n}\bar{w}_{n}}{1+\bar{w}_{n}}=c\bar{v}_{n},\ \
\frac{k\bar{v}_{n}}{1+\bar{w}_{n}}=q\bar{w}_{n},\ \
\underline{u}_{n}=\frac{\theta(1+\bar{w}_{n})}{a+a\bar{w}_{n}+b\bar{w}_{n}},$
$\displaystyle\displaystyle\frac{b\underline{u}_{n}\underline{w}_{n}}{1+\underline{w}_{n}}=c\underline{v}_{n},\
\ \frac{k\underline{v}_{n}}{1+\underline{w}_{n}}=q\underline{w}_{n},\ \
\bar{u}_{n+1}=\frac{\theta(1+\underline{w}_{n})}{a+a\underline{w}_{n}+b\underline{w}_{n}}.$

Consequently, $\underline{u}_{\infty}$, $\bar{u}_{\infty}$,
$\underline{v}_{\infty}$, $\bar{v}_{\infty}$, $\underline{w}_{\infty}$,
$\bar{w}_{\infty}$ satisfy

$\displaystyle\displaystyle\frac{b\bar{u}_{\infty}\bar{w}_{\infty}}{1+\bar{w}_{\infty}}=c\bar{v}_{\infty},\
\ \frac{k\bar{v}_{\infty}}{1+\bar{w}_{\infty}}=q\bar{w}_{\infty},\ \
\underline{u}_{\infty}=\frac{\theta(1+\bar{w}_{\infty})}{a+a\bar{w}_{\infty}+b\bar{w}_{\infty}},$
$\displaystyle\displaystyle\frac{b\underline{u}_{\infty}\underline{w}_{\infty}}{1+\underline{w}_{\infty}}=c\underline{v}_{\infty},\
\
\frac{k\underline{v}_{\infty}}{1+\underline{w}_{\infty}}=q\underline{w}_{\infty},\
\
\bar{u}_{\infty}=\frac{\theta(1+\underline{w}_{\infty})}{a+a\underline{w}_{\infty}+b\underline{w}_{\infty}}.$

Using our assumptions $\mathcal{R}_{0}>1$ and $b/a\leq 2$, we can derive after
a series of calculations that

$\underline{u}_{\infty}=\bar{u}_{\infty}=u^{*},\ \
\underline{v}_{\infty}=\bar{v}_{\infty}=v^{*},\ \
\underline{w}_{\infty}=\bar{w}_{\infty}=w^{*}.$

Thus (4.11) holds and the proof is ended. ∎

##  5 Criteria for spreading and vanishing

In this section we study the criteria governing spreading
($h_{\infty}-g_{\infty}=\infty$) and vanishing
($h_{\infty}-g_{\infty}<\infty$). In the following, we divide our discussion
into two cases based on the Basic Reproduction Number ${\cal R}_{0}=\theta
kb/(acq)$. For convenience, we denote $\gamma=\max\left\\{\mu,\beta\right\\}$.

###  5.1 The case ${\cal R}_{0}\leq 1$

######  Theorem 5.1.

Let $(u,v,w,g,h)$ be the unique solution of (1.2). If ${\cal R}_{0}\leq 1$,
then $h_{\infty}-g_{\infty}<\infty$.

###### Proof.

By a simple comparison argument, we have

$u(t,x)\leq{\theta}/{a}+\|u_{0}\|_{\infty}e^{-at}=:\hat{u}(t)\ \ {\rm
for~{}}t\geq 0,\ x\in\mathbb{R}.$

Hence $v$ satisfies

$\displaystyle\left\\{\begin{aligned} &v_{t}\leq
f_{2}(\hat{u}(t),\,v,w),&&t>0,~{}g(t)<x<h(t),\\\
&v(t,g(t))=v(t,h(t))=0,&&t>0,\\\ &v(0,x)=v_{0}(x),&&|x|\leq
h_{0}.\end{aligned}\right.$

Notice that ${\cal R}_{0}=\theta kb/acq\leq 1$. It follows that by simple
calculations

$\displaystyle\displaystyle\frac{{\rm d}}{{\rm
dt}}\int_{g(t)}^{h(t)}\big{(}cw+kv\big{)}{\rm d}x$ $\displaystyle=$
$\displaystyle\int_{g(t)}^{h(t)}\big{(}cw_{t}+kv_{t}\big{)}{\rm d}x$
$\displaystyle\leq$
$\displaystyle\int_{g(t)}^{h(t)}\left[cdw_{xx}+kb\|u_{0}\|_{\infty}e^{-at}w+(k\theta
b/a-qc)w\right]{\rm d}x$ $\displaystyle\leq$
$\displaystyle\int_{g(t)}^{h(t)}\left(cdw_{xx}+kb\|u_{0}\|_{\infty}e^{-at}w\right){\rm
d}x$ $\displaystyle\leq$ $\displaystyle-
cd\gamma^{-1}(h^{\prime}(t)-g^{\prime}(t))+kbA_{3}\|u_{0}\|_{\infty}e^{-at}(h(t)-g(t)).$

Set

$f(t)=\int_{g(t)}^{h(t)}\big{(}cw+kv\big{)}{\rm d}x,\ \ \ell(t)=h(t)-g(t),\ \
\varphi(t)=kbA_{3}\|u_{0}\|_{\infty}e^{-at}.$

Then we have

$cd\ell^{\prime}(t)\leq-\gamma f^{\prime}(t)+\gamma\varphi(t)\ell(t).$

Integrating the above differential inequality from $0$ to $t$ yields

$\ell(t)\leq\ell(0)+{\gamma}(cd)^{-1}f(0)+{\gamma}(cd)^{-1}\int_{0}^{t}\varphi(s)\ell(s){\rm
d}s.$

By virtue of the Gronwall inequality,

$\ell(t)\leq\big{[}\ell(0)+{\gamma}(cd)^{-1}f(0)\big{]}\exp\left\\{{{\gamma}(cd)^{-1}\int_{0}^{t}\varphi(s){\rm
d}s}\right\\}<\infty.$

Thus, $h_{\infty}-g_{\infty}<\infty$.∎

###  5.2 The case ${\cal R}_{0}>1$

In this subsection, we always assume that ${\cal R}_{0}>1$, and consider $d$,
$h_{0}$, $\mu$ and $\beta$ as varying parameters to depict the criteria for
spreading and vanishing.

######  Theorem 5.2.

Let $(u,v,w,g,h)$ be the solution of (1.2). If $h_{\infty}-g_{\infty}<\infty$,
then we have

$h_{\infty}-g_{\infty}\leq\pi\sqrt{acd/(kb\theta-acq)}=:\Lambda.$

This implies that if $h_{0}\geq\Lambda/2$, then
$h_{\infty}-g_{\infty}=\infty$. Moreover if $h_{\infty}-g_{\infty}=\infty$,
then

$\displaystyle\displaystyle\limsup_{t\to\infty}\|v(t,\cdot)\|_{C([g(t),h(t)])}>0,\
\ \displaystyle\limsup_{t\to\infty}\|w(t,\cdot)\|_{C([g(t),h(t)])}>0.$ (5.1)

###### Proof.

Due to Theorem 4.1 and $h_{\infty}-g_{\infty}<\infty$, we see that
$\displaystyle\lim_{t\to\infty}u(t,\cdot)=\theta/{a}$ uniformly in
$\mathbb{R}$, and

$\displaystyle\lim_{t\to\infty}\|v(t,\cdot)\|_{C([g(t),h(t)])}=0,\ ~{}\
~{}\lim_{t\to\infty}\|w(t,\cdot)\|_{C([g(t),h(t)])}=0.$ (5.2)

Arguing indirectly, if $h_{\infty}-g_{\infty}>\Lambda$, then there exists
$T>0$ such that for any small $\varepsilon>0$ satisfying
$\frac{kb}{c}(\frac{\theta}{a}-\varepsilon)-q>0$, we have

$\displaystyle u(t,x)>{\theta}/{a}-\varepsilon,\ \ \forall\ t\geq T,\
x\in\mathbb{R};$ $\displaystyle
h(t)-g(t)>\pi\sqrt{d}\left[kb({\theta}/{a}-\varepsilon)/c-q\right]^{-1/2}=:\Lambda_{\varepsilon},\
\ \forall\ t\geq T.$

Then for any $[l_{1},l_{2}]\subseteq(g(T),h(T))$ and
$l_{2}-l_{1}>\Lambda_{\varepsilon}$, we have

$\displaystyle\left\\{\begin{aligned} &v_{t}\geq
f_{2}({\theta}/{a}-\varepsilon,\,v,w),&&t>T,\ \ l_{1}<x<l_{2},\\\
&w_{t}-dw_{xx}=f_{3}(v,w),&&t>T,\ \ l_{1}<x<l_{2},\\\ &w(t,x)>0,&&t>T,\
x=l_{i},\ i=1,2,\\\ &v(T,x)>0,\ w(T,x)>0,&&l_{1}\leq x\leq
l_{2}.\end{aligned}\right.$ (5.3)

Consider the following eigenvalue problem

$\displaystyle\left\\{\begin{aligned}
&d\phi_{xx}-q\phi+k\psi=\lambda\phi,&&l_{1}<x<l_{2},\\\
&b(\theta/{a}-\varepsilon)\phi-c\psi=\lambda\psi,&&l_{1}<x<l_{2},\\\
&\phi(l_{i})=0,&&i=1,2.\end{aligned}\right.$ (5.4)

Denote the principal eigenpair of (5.4) by $(\lambda_{1},(\phi,\psi))$ with
$\max_{x\in[l_{1},l_{2}]}|\phi(x)|=1$. It follows from Corollary 3.2 that
$\lambda_{1}>0$ due to $l_{2}-l_{1}>\Lambda_{\varepsilon}$. Let

$\underline{v}(x)=\delta\psi(x),\ \ \ \underline{w}(x)=\delta\phi(x)$

with $\delta>0$ to be determined later.

We claim that there exists $\delta_{0}>0$ sufficiently small such that for any
$0<\delta<\delta_{0}$, we have

$\displaystyle\left\\{\begin{aligned}
&0<f_{2}(\theta/a-\varepsilon,\,\underline{v},\underline{w}),\ \
l_{1}<x<l_{2},\\\ &-d\underline{w}_{xx}\leq
f_{3}(\underline{v},\underline{w}),\ \ l_{1}<x<l_{2}.\end{aligned}\right.$
(5.5)

In fact, we have

$\displaystyle
f_{2}(\theta/a-\varepsilon,\,\underline{v},\underline{w})=\delta\frac{b(\theta/a-\varepsilon)\phi}{1+\delta\phi}-c\delta\psi=\delta\frac{(c+\lambda_{1})\psi}{1+\delta\phi}-c\delta\psi=\delta\left(\frac{c+\lambda_{1}}{1+\delta\phi}-c\right)\psi>0$

provided that $\delta>0$ is small. The proof of the second inequality of (5.5)
can be done in a similar manner.

Furthermore, one can choose small $\delta>0$ such that
$v(T,x)\geq\delta\psi(x)$ and $w(T,x)\geq\delta\phi(x)$ for
$x\in[l_{1},l_{2}]$. Then $(\underline{v},\underline{w})$ satisfies

$\displaystyle\left\\{\begin{aligned} &\underline{v}_{t}\leq
f_{2}(\theta/a-\varepsilon,\,\underline{v},\underline{w}),&&t>T,\ \
l_{1}<x<l_{2},\\\ &\underline{w}_{t}-d\underline{w}_{xx}\leq
f_{3}(\underline{v},\underline{w}),&&t>T,\ \ l_{1}<x<l_{2},\\\
&\underline{v}(t,x)=0,\ \underline{w}(t,x)=0,&&t>T,\ x=l_{i},\ i=1,2,\\\
&\underline{v}(T,x)\leq v(T,x),\ \underline{w}(T,x)\leq w(T,x),&&l_{1}\leq
x\leq l_{2}.\end{aligned}\right.$

By virtue of the comparison principle,

$\displaystyle v(t,x)\geq\underline{v}(x),\ \ \ w(t,x)\geq\underline{w}(x),\ \
t\geq T,\ l_{1}\leq x\leq l_{2}.$ (5.6)

This is a contradiction with (5.2).

We now assume $h_{\infty}-g_{\infty}=\infty$ and prove (5.1). By the
comparison principle, it is easy to see that
$\lim\limits_{t\to\infty}\|v(t,\cdot)\|_{C([g(t),h(t)])}=0$ if and only if
$\lim\limits_{t\to\infty}\|w(t,\cdot)\|_{C([g(t),h(t)])}=0$. Hence if we
assume that one of the two limits in (5.1) does not hold, we can similarly
obtain $\displaystyle\lim_{t\to\infty}u(t,\cdot)=\theta/{a}$ uniformly in
$\mathbb{R}$. By $h_{\infty}-g_{\infty}=\infty$, we can derive an analogous
contradiction as above. The proof is ended. ∎

Obviously, $h_{0}\geq\Lambda/2$ is equivalent to $d\leq 4h_{0}^{2}q({\cal
R}_{0}-1)\pi^{-2}=:D$. So the above result suggests that when ${\cal
R}_{0}>1$, the larger initial habitat $[-h_{0},h_{0}]$ or the lower dispersal
rate $d$ of the virus is, the more possibility of successful spreading is
observed.

######  Theorem 5.3.

If $h_{\infty}-g_{\infty}=\infty$, then $h_{\infty}=\infty$ and
$g_{\infty}=-\infty$.

###### Proof.

By way of contradiction, we assume that $h_{\infty}<\infty$ and
$g_{\infty}=-\infty$. If we take $L>\Lambda+2$, where $\Lambda$ is defined in
Theorem 5.2, then there exists $T_{0}>0$ such that $g(T_{0})<-L$. Then $w$
satisfies

$\displaystyle\left\\{\begin{aligned} &w_{t}-dw_{xx}=f_{3}(v,w),&&t>T_{0},\ \
-L<x<h(t),\\\ &w(t,-L)>0,\ w(t,h(t))=0,&&t>T_{0},\\\ &h^{\prime}=-\beta
w_{x}(t,h(t)),&&t>T_{0},\\\ &w(T_{0},x)\geq 0,&&-L\leq x\leq
h(T_{0}).\end{aligned}\right.$

As $h_{\infty}<\infty$, using the second estimate in (2.7) we have
$\displaystyle\lim_{t\rightarrow\infty}h^{\prime}(t)=0$. Then, using the first
estimate in (2.7) and Lemma 3.3, one can arrive at

$\displaystyle\lim\limits_{t\to\infty}\|w(t,\cdot)\|_{C([-L,\,h(t)])}=0.$
(5.7)

Then we may argue as in the proof of Theorem 4.1 with minor modifications to
derive that

$\lim_{t\to\infty}\max_{[1-L,\,h(T_{0})]}v(t,\cdot)=0,\ \
\lim_{t\to\infty}\max_{[1-L,\,h(T_{0})]}u(t,\cdot)={\theta}/{a}.$

There exists $T>T_{0}$ such that $u(t,x)\geq{\theta}/{a}-\varepsilon$ for
$(t,x)\in[T,\infty)\times[1-L,\,h(T_{0})]$. Let $\varepsilon>0$ be small
enough satisfying $L-1>\Lambda_{\varepsilon}$, where $\Lambda_{\varepsilon}$
is defined as in Theorem 5.2. Choose an interval
$[l_{1},l_{2}]\subset(1-L,\,h(T_{0}))$ with
$l_{2}-l_{1}>\Lambda_{\varepsilon}$. Then $(v,w)$ satisfies (5.3). As in the
proof of Theorem 5.2, we can conclude that (5.6) holds. This is a
contradiction with (5.7).

Analogously, we can prove that the case with $h_{\infty}=\infty$ and
$g_{\infty}>-\infty$ also does not hold. Therefore, we must have
$h_{\infty}=\infty$ and $g_{\infty}=-\infty$.∎

The following result implies that although the initial habitat is small or the
dispersal rate is fast, the spreading also can occur if the expanding rate
$\mu$ or $\beta$ is appropriately large. By using similar method in the proof
of [34, Lemma 3.2] with some modifications, we can prove the following lemma.

######  Lemma 5.4.

If $h_{0}<\Lambda/2$ $($or $d>D)$, then there exists $\mu^{0}>0$ $(resp.\
\beta^{0}>0)$ such that if $\mu\geq\mu^{0}$ $(resp.\ \beta\geq\beta^{0})$,
then $h_{\infty}-g_{\infty}=\infty$.

The above lemma also indicates that if
$\gamma=\max\left\\{\mu,\beta\right\\}\geq\max\\{\mu^{0},\beta^{0}\\}$, then
$h_{\infty}-g_{\infty}=\infty$. Instinctively, we deem that if ${\cal
R}_{0}>1$, $h_{0}<\Lambda/2$ (or $d>D$), $\mu$ and $\beta$ both are small,
then the vanishing will happen. The lemma listed below supports our belief.

######  Lemma 5.5.

Assume $h_{0}<\Lambda/2$ $($or $d>D)$. Then there exists $\mu_{0}>0$ such that
when $\gamma=\max\left\\{\mu,\beta\right\\}\leq\mu_{0}$, we must have
$h_{\infty}-g_{\infty}<\infty$.

###### Proof.

Let $\hat{u}$ be the unique solution of the problem

$\displaystyle\hat{u}_{t}=\theta-a\hat{u},\ \ t>0;\ \ \
\hat{u}(0)=\max\\{\|u_{0}\|_{\infty},\,\theta/a\\}.$

Then $\hat{u}(t)\geq\theta/a$ and
$\displaystyle\lim_{t\to\infty}\hat{u}(t)=\theta/a$. By the comparison
principle, $u(t,x)\leq\hat{u}(t)$ in $[0,\infty)\times\mathbb{R}$. For any
fixed $h_{0}<l<\Lambda/2$, we consider the following eigenvalue problem

$\displaystyle\left\\{\begin{aligned}
&d\phi_{xx}-q\phi+k\psi=\lambda\phi,&&-l<x<l,\\\
&(b\theta/a)\phi-c\psi=\lambda\psi,&&-l<x<l,\\\ &\phi(\pm
l)=0.\end{aligned}\right.$

In view of Corollary 3.2, the principal eigenvalue $\lambda_{1}<0$ since
$2l<\Lambda$. Moreover, by (3.2), there exists a positive constant
$\tilde{\phi}$ such that

$\displaystyle-d\displaystyle\frac{\pi^{2}}{(2l)^{2}}\tilde{\phi}-q\tilde{\phi}+k=\lambda_{1}\tilde{\phi},\
\ \ \frac{b\theta}{a}\tilde{\phi}-c=\lambda_{1}.$ (5.8)

Define

$\displaystyle
f(t)=M\exp\left\\{\displaystyle\int_{0}^{t}\left[\tilde{\phi}b(\hat{u}(s)-{\theta}/{a})+\lambda_{1}\right]{\rm
d}s\right\\},$ $\displaystyle
r(t)=\left(h^{2}_{0}+\gamma\pi\tilde{\phi}\displaystyle\int_{0}^{t}f(s){\rm
d}s\right)^{1/2},$ $\displaystyle\hat{v}(t,x)=f(t)\cos\displaystyle\frac{\pi
x}{2r(t)},\ \ \hat{w}(t,x)=\tilde{\phi}f(t)\cos\displaystyle\frac{\pi
x}{2r(t)},\ \ -r(t)\leq x\leq r(t),$

where $\gamma=\max\left\\{\mu,\beta\right\\}$, and $M>0$ is taken large so
that

$\displaystyle v_{0}(x)\leq M\cos\displaystyle\frac{\pi
x}{2h_{0}}=\hat{v}(t,0),\ \
w_{0}(x)\leq\tilde{\phi}M\cos\displaystyle\frac{\pi x}{2h_{0}}=\hat{w}(t,0)\ \
\mbox{in}\ \ [-h_{0},h_{0}].$

As $\displaystyle\lim_{t\to\infty}\hat{u}(t)=\theta/a$ and $\lambda_{1}<0$, it
follows that $\int_{0}^{\infty}f(s){\rm d}s<\infty$. Clearly,
$r^{\prime}(t)>0$ for $t\geq 0$. Set

$\mu_{0}=\displaystyle\frac{l^{2}-h^{2}_{0}}{\pi\tilde{\phi}\int_{0}^{\infty}f(s){\rm
d}s}.$

Then $r(t)<l$ for $t\geq 0$ provided $0<\gamma\leq\mu_{0}$.

Using (5.8) and $\hat{u}(t)\geq\theta/a$, $r(t)<l$ for all $t\geq 0$, by a
series of calculations we have

$\displaystyle\hat{v}_{t}-f_{2}(\hat{u},\hat{v},\hat{w})$ $\displaystyle\geq$
$\displaystyle f(t)\cos\displaystyle\frac{\pi
x}{2r(t)}\left(\lambda_{1}+c-\displaystyle\frac{\theta}{a}b\tilde{\phi}\right)=0,$
$\displaystyle\hat{w}_{t}-d\hat{w}_{xx}-f_{3}(\hat{v},\hat{w})$
$\displaystyle\geq$ $\displaystyle f(t)\cos\displaystyle\frac{\pi
x}{2r(t)}\left[\tilde{\phi}^{2}b\left(\hat{u}-\frac{\theta}{a}\right)+d\tilde{\phi}\left(\displaystyle\frac{\pi}{2r(t)}\right)^{2}+\lambda_{1}\tilde{\phi}+q\tilde{\phi}-k\right]$
$\displaystyle=$ $\displaystyle\tilde{\phi}f(t)\cos\displaystyle\frac{\pi
x}{2r(t)}\left[\tilde{\phi}b\left(\hat{u}-\frac{\theta}{a}\right)+d\left(\displaystyle\frac{\pi}{2r(t)}\right)^{2}-d\left(\displaystyle\frac{\pi}{2l}\right)^{2}\right]\geq
0$

for $t>0$ and $-r(t)<x<r(t)$. And we easily see

$-r^{\prime}(t)=-\gamma\hat{w}_{x}(t,-r(t)),\ \
r^{\prime}(t)=-\gamma\hat{w}_{x}(t,r(t)).$

Thus for any $0<\gamma\leq\mu_{0}$, $(\hat{u},\hat{v},\hat{w},-r,r)$ satisfies
$r(0)=h_{0}$ and

$\displaystyle\left\\{\begin{aligned} &\hat{u}_{t}=\theta-a\hat{u},&&t>0,\\\
&\hat{v}_{t}\geq f_{2}(\hat{u},\hat{v},\hat{w}),&&t>0,~{}-r(t)<x<r(t),\\\
&\hat{w}_{t}-d\hat{w}_{xx}\geq
f_{3}(\hat{v},\hat{w}),&&t>0,~{}-r(t)<x<r(t),\\\ &\hat{v}(t,\pm
r(t))=\hat{w}(t,\pm r(t))=0,&&t>0,\\\
&-r^{\prime}(t)\leq-\mu\hat{w}_{x}(t,-r(t)),\
r^{\prime}(t)\geq-\beta\hat{w}_{x}(t,r(t)),&&t\geq 0,\\\ &\hat{u}(0)\geq
u_{0}(x),&&-\infty<x<\infty,\\\ &\hat{v}(0,x)\geq v_{0}(x),~{}\hat{w}(0,x)\geq
w_{0}(x),&&|x|\leq h_{0}.\end{aligned}\right.$

By the comparison principle (Lemma 3.4), $-r(t)\leq g(t)$, $h(t)\leq r(t)$ for
$t\geq 0$. As a result, we have

$g_{\infty}\geq-\lim_{t\to\infty}r(t)\geq-l,\ ~{}\
~{}h_{\infty}\leq\lim_{t\to\infty}r(t)\leq l,$

which implies $h_{\infty}-g_{\infty}<\infty$. This completes the proof. ∎

According to the above proof, we can see that $\mu_{0}$ is independent of
$v_{0}$ and $w_{0}$ and strictly decreasing in $M$. Thus for any given $\mu$
and $\beta$, there exists $M>0$ sufficiently small such that
$\gamma\leq\mu_{0}$. Meanwhile, if both $v_{0}$ and $w_{0}$ are small enough
such that for such $M$

$\displaystyle v_{0}(x)\leq M\cos\displaystyle\frac{\pi x}{2h_{0}},\ \
w_{0}(x)\leq\tilde{\phi}M\cos\displaystyle\frac{\pi x}{2h_{0}},\ \ \forall\
x\in[-h_{0},h_{0}],$

we still can derive $h_{\infty}-g_{\infty}<\infty$ by the above arguments.
Hence we have the following conclusion.

######  Remark 5.6.

Assume $h_{0}<\Lambda/2\,(d>D)$, and that $(v_{0},w_{0})$ satisfies (1.3).
Then vanishing happens if both $v_{0}$ and $w_{0}$ are small enough.

Combining the above two lemmas, by the similar arguments in [28, Theorem 5.2]
we can show the following conclusion.

######  Theorem 5.7.

If $h_{0}<\Lambda/2\,(d>D)$. There exists $0<\mu_{*}\leq\mu^{*}$ such that
$h_{\infty}-g_{\infty}<\infty$ if $\gamma\leq\mu_{*}$ or $\gamma=\mu^{*}$, and
$h_{\infty}-g_{\infty}=\infty$ if $\gamma>\mu^{*}$.

##  6 Discussion

In this paper we proposed a viral propagation model with nonlinear infection
rate and free boundaries and investigated the dynamical properties. This model
is composed of two ordinary differential equations and one partial
differential equation, in which the spatial range of the first equation is the
whole space $\mathbb{R}$, and the last two equations have free boundaries. As
a new mathematical model, we have proved the existence, uniqueness and uniform
estimates of global solution, and provided the criteria for spreading and
vanishing, and long time behavior of the solution components $u,v,w$.

Comparing with the corresponding ordinary differential systems, the Basic
Reproduction Number ${\cal R}_{0}=\theta kb/(acq)$ plays a different role:

  * (i)

For the corresponding ordinary differential systems, by the Lyapunov function
method we can prove that if ${\cal R}_{0}<1$ then the infection can not spread
successfully, while if ${\cal R}_{0}>1$ then the infection will spread
successfully. When ${\cal R}_{0}=1$ the dynamical property is not clear;

  * (ii)

For our present model, the results indicate that when ${\cal R}_{0}\leq 1$,
the virus cannot spread successfully; when ${\cal R}_{0}>1$, the successful
spread of virus depends on the initial value and varying parameters. If the
initial occupying area $[-h_{0},h_{0}]$ is beyond a critical size, namely
$2h_{0}\geq\pi\sqrt{acd/(bk\theta-acq)}$, then spreading happens regardless of
the moving parameter $\mu$, $\beta$ and initial population density
$(u_{0},v_{0},w_{0})$. While $2h_{0}<\pi\sqrt{acd/(bk\theta-acq)}$, whether
spreading or vanishing happens depends on the initial population density
$(v_{0},w_{0})$ and the moving parameter $\mu$ and $\beta$.

From a biological point of view, our model and results seem closer to the
reality. On the other hand, our model shows more complex and precise dynamical
properties from a mathematical point of view.

Before ending this paper, we mention that for the corresponding Cauchy
problem:

$\displaystyle\begin{cases}u_{t}-d_{1}\Delta
u=f_{1}(u,w),&x\in\mathbb{R}^{n},\ t>0,\\\ v_{t}-d_{2}\Delta
v=f_{2}(u,v,w),&x\in\mathbb{R}^{n},\ t>0,\\\ w_{t}-d_{3}\Delta
w=f_{3}(v,w),&x\in\mathbb{R}^{n},\ t>0,\\\ u(0,x)=u_{0}(x),\ v(0,x)=v_{0}(x),\
w(0,x)=w_{0}(x),&x\in\mathbb{R}^{n},\end{cases}$

we guess that the Basic Reproduction Number is still ${\cal R}_{0}=\theta
kb/(acq)$ and it plays the same role as in the corresponding ODEs.

Acknowledgments: The authors would like to thank the reviewers for their
helpful comments and suggestions that significantly improve the initial
version of this paper.

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