#  A nonlocal diffusion single population model in advective environment

Yaobin Tang, Binxiang Dai111Corresponding author. Email address:
<EMAIL_ADDRESS>(B. Dai).
School of Mathematics and Statistics, HNP-LAMA,
Central South University,
Changsha, Hunan 410083, P. R. China

###### Abstract

This paper is devoted to a nonlocal reaction-diffusion-advection model that
describes the spatial dynamics of freshwater organisms in a river with a
directional motion. Our goal is to investigate how the advection rate affects
the dynamic behaviors of species. We first establish the well-posedness of
global solutions, where the regularized problem containing a viscosity term
and the re-established maximum principle play an important role. And we then
show the existence/nonexistence, uniqueness, and stability of nontrivial
stationary solutions by analyzing the principal eigenvalue of integro-
differential operator (especially studying the monotonicity of the principal
eigenvalue with respect to the advection rate), which enables us to understand
the longtime behaviors of solutions and obtain the sharp criteria for
persistence or extinction. Furthermore, we study the limiting behaviors of
solutions with respect to the advection rate and find that the sufficiently
large directional motion will cause species extinction in all situations.
Lastly, the numerical simulations verify our theoretical proofs.

Keywords: Nonlocal diffusion; Advection; Nontrivial stationary solution;
Dirichlet boundary; Neumann boundary

MSC: 35K57, 35R35, 35B40, 92D25

##  1 Introduction

The dispersal ways of populations have received much attention recently and
are a major focus of most scholars. The reaction-diffusion equation is a
mathematical model commonly used to describe the spatial random diffusion and
evolution of biological populations or diseases [1, 2, 3].

Hutson and Martinez [4] found that the model, derived from random diffusion,
is too restrictive to model the seed dispersal of a single species with long-
range dispersal. And they thought that the position-jump process is more
suitable. Let $u(t,x)$ be the population density at time $t$ and location $x$.
Assume that the population $u(t,x)$ can jump from one position $x$ to another
position $y$ with a certain probability $J(y-x)$. Then
$\int_{\mathbb{R}}J(x-y)u(t,y)dy$ represents the population density arriving
to position $x$ from all other positions and $\int_{\mathbb{R}}J(y-x)u(t,x)dy$
denotes the population density leaving position $x$ to all other positions.
Hence, the nonlocal diffusion, described by the convolution operator

$(J*u)(t,x)=d\int_{\mathbb{R}}J(x-y)u(t,y)dy-du(t,x),$

can be introduced to model the dispersion of individuals [5, 6, 7], where $d$
is the dispersal rate and the kernel function $J:\mathbb{R}\to\mathbb{R}^{+}$
is continuous, nonnegative, and has the properties

  1. (J):

$J(0)>0,\int_{\mathbb{R}}J(x)dx=1$.

Denote $\Omega\subset\mathbb{R}$ is a smooth domain. If the dispersal takes
place in $\mathbb{R}$ and species $u$ vanishes outside $\Omega$, then

$\begin{cases}u_{t}(t,x)=d\int_{\Omega}J(x-y)u(t,y)dy-
du(t,x),&t>0,x\in{\Omega},\\\
u(t,x)=0,&t>0,x\in\mathbb{R}\backslash{\Omega}\end{cases}$

is called the nonlocal diffusion equation with homogeneous Dirichlet boundary
condition [8, 9]. Biologically, the region $\Omega$ outside is considered a
hostile environment, and any individual that jumps outside of $\Omega$ dies
instantly. Note that the boundary condition is not understood in the usual
sense, as we are not imposing that $u|_{\partial\Omega}=0$.

If it takes into account the diffusion inside $\Omega$, as we have explained,
no individual enters or leaves the domain $\Omega$, then

$u_{t}(t,x)=d\int_{\Omega}J(x-y)[u(t,y)-u(t,x)]dy,t>0,x\in{\Omega},$

is called the nonlocal diffusion equation with classical Neumann boundary
condition in the literature [10, 11].

There are a lot of researches on nonlocal diffusion problems in
biomathematics. In recent years, the following nonlocal evolution equation

$\begin{cases}u_{t}(t,x)=d\int_{\Omega}J(x-y)u(t,y)dy-
du(t,x)+f(x,u),&t>0,x\in\Omega,\\\ u(0,x)=\phi(x),&x\in\Omega\end{cases}$
(1.1)

has been considered. When $\Omega=\mathbb{R}^{N}$, several studies focused on
the analysis of entire solutions, propagation speeds, and the corresponding
traveling wave problems of (1.1), see [12, 13, 14, 15, 16, 17]. When
$\Omega\subset\mathbb{R}^{N}$ is bounded, there are some works focusing on the
model with Dirichlet/Neumann boundary condition, see [18, 19, 20, 21].
Moreover, the nonlocal problem was applied in various specific situations,
such as the studies of asymptotic behaviors [22, 7, 23, 24], spectral theory
[20, 21, 25], application in infectious disease models [26, 27], and the
analysis of systems with Lotka-Volterra type [28, 29, 30].

In addition, some researchers are devoted to the nonlocal problem with a
Fisher-KPP type, bistable type or general nonlinear reaction term.
Specifically, Bates & Zhao [18] investigated the spatial dynamics of organisms
or cells. To be specific, they considered the following model:

$\begin{cases}u_{t}(t,x)=d\int_{\Omega}J(x-y)u(t,y)dy+b(x)u(t,x)+f(x,u),&t>0,x\in\Omega,\\\
u(0,x)=\phi(x),&x\in\Omega,\end{cases}$

where $f(x,u)\in C^{1}(\bar{\Omega}\times\mathbb{R}^{+})$ is strictly
sublinear with respect to $u$ satisfying $f(x,0)\equiv 0$. Subsequently, Sun
[31] discussed the nonlocal problem (1.1) with reaction term $f(x,u)=\lambda
g(x)h(u)$, which arises from selection-migration models in genetics. Here
$g\in C(\bar{\Omega})$ is sign changed and $h:[0,1]\to\mathbb{R}^{+}$ is a
smooth function such that $h(0)=h(1)=0$ and $h^{\prime\prime}(u)<0$ for $0\leq
u\leq 1$. Moreover, [32] focused on the nonlocal problem (1.1) with a KPP-type
reaction term $f(x,u)\in C^{1,\alpha}(\mathbb{R}^{2})$ which satisfies

$\limsup_{|x|\to\infty}\frac{f(x,s)}{s}<0\ \ \mbox{uniformly in}\ s\geq 0.$

Nextly, the nonlocal problem (1.1) with $f(x,u)=r(x)u(1-\frac{u}{K(x)})$ was
studied by Su [33], where $r(x),K(x)>0$ on $\bar{\Omega}$ denote respectively
its intrinsic growth rate and its carrying capacity. And the nonlocal problem
(1.1) with $f(x,u)=m(x)u-c(x)u^{p}$ was analyzed by Sun [34], where $m(x)$
stands for the intrinsic growth rate and is sign changed, $c(x)>0$ on
$\bar{\Omega}$ measures the capacity of $\Omega$ to support the species. They
provided the existence, uniqueness, and stability of positive stationary
solutions, and obtained the longtime behaviors through certain threshold
values. It is worth mentioning that, the selection of threshold value usually
depends on the sign of the principal eigenvalue of the operator:

$\displaystyle(\mathfrak{L}_{\Omega,J}+a)[\varphi(x)]:=d\int_{\Omega}J(x-y)\varphi(y)dy-d\varphi(x)+a(x)\varphi(x),$

which is defined on $C^{1}(\bar{\Omega})$ and $a(x)\in C(\bar{\Omega})\cap
L^{\infty}(\Omega)$. It is a crucial factor in determining the dynamic
behaviors of the solution.

In addition to the spatial diffusion of populations, various environmental
factors such as wind direction, river flow, climate, and resource distribution
can influence the longtime dynamic behaviors of biological populations and
diseases. To account for this phenomenon, researchers include an advection
term to describe the directed movement of the population or disease over
space.

In 1954, Muller [35] observed that organisms living in streams or rivers are
constantly washed downstream or out of the original ecological environment,
resulting in a general decline in species, even extinction. But some species
can survive in this environment from generation to generation, and this
biological phenomenon often is called ”drift paradox”. So they puzzled over a
question: How do species in rivers survive without being washed away? Then,
Speirs & Gurney introduced the reaction-advection-diffusion equation in [36]
to describe the dynamic behaviors of single species under the influence of
currents:

$\begin{cases}u_{t}(t,x)=du_{xx}(t,x)-qu_{x}(t,x)+uf(u),&t>0,x\in(0,L),\\\
du_{x}(t,0)-qu(t,0)=0,u(t,L)=0,&t>0,\\\ u(0,x)=\phi(x),&x\in(0,L).\end{cases}$

The biological explanation for this is that the population $u$ not only
undergoes random diffusion but also experiences advective transport at
velocity $q$. They demonstrated that a large enough random diffusion can
counterbalance the directional motion caused by currents, leading to the
survival of species. Subsequently, the persistence of single or multiple
populations in advective environments was studied [37, 38, 39, 40, 41]. Some
significant and profound results have been obtained for reaction-diffusion
problems in advective environments. However, the question remains whether
these results can be extended to nonlocal diffusion problems in advective
environments. As far as the author knows, in [42], the monotonicity and
uniqueness of positive solutions of integro-differential equation

$\int_{\mathbb{R}}J(x-y)u(y)dy-u(x)-cu^{\prime}(x)+f(u)=0,x\in\mathbb{R}$

was studied. Its positive solution can be regarded as the traveling wave
solution $u(t,x)=\phi(x+ct)$ for the following nonlocal problem:

$u_{t}(t,x)=\int_{\mathbb{R}}J(x-y)u(t,y)dy-u(t,x)+f(u),\
(t,x)\in\mathbb{R}^{+}\times\mathbb{R}.$

Then, Zhao & Shen et al. [43, 44] discussed a nonlocal reaction-diffusion
equation in shifting habitat:

$u_{t}(t,x)=\int_{\mathbb{R}}J(x-y)u(t,y)dy-u(t,x)+f(x-ct,u)u,(t,x)\in\mathbb{R}^{+}\times\mathbb{R}.$

Biologically, it represents that the population will change with the moving
habitat of speed $c$ due to climate change. Considering the function
$v(t,\zeta)$ that satisfies the form $u(t,x)=v(t,x-ct)$ with $v(t,x)$ being
differentiable and $\zeta=x-ct$, we find that $v(t,\zeta)$ satisfies

$\displaystyle
u_{t}(t,\zeta)=cu_{\zeta}(t,\zeta)+\int_{\mathbb{R}}J(\zeta-y)u(t,y)dy-u(t,\zeta)+f(\zeta,u)u,(t,\zeta)\in\mathbb{R}^{+}\times\mathbb{R}.$
(1.2)

And then, the nonexistence/existence of nontrivial stationary solutions of
(1.2) was obtained by studying the spectral theory of integro-differential
operator:

$(\mathfrak{L}_{\Omega,J,q}+a)[\varphi(x)]:=d\int_{\Omega}J(x-y)\varphi(y)dy-q\varphi^{\prime}(x)+a(x)\varphi(x),$

which was studied by [20, 32, 44, 45].

In addition, observe that [46] considered fresh water organisms living in a
river domain ${\Omega}=(l_{1},l_{2})$. Assume that these organisms not only
can move within the river due to random diffusion, but are also subject to
directed movement arising from currents. Concretely, these organisms are
affected by the velocity of currents from left to right and they can not
survive on land $(-\infty,l_{1})$ and in the ocean $(l_{2},+\infty)$:

$\begin{cases}u_{t}=d[\int_{l_{1}}^{l_{2}}J(x-y)u(t,y)dy-u(t,x)]-(c(x)u(t,x))_{x},&t>0,x\in(l_{1},l_{2}),\\\
u(t,l_{1})=0,&t>0,\end{cases}$

where $c(x)>0$ on $[l_{1},l_{2}]$ is the velocity function of water. They
emphasized the importance of boundary condition $u(t,l_{1})=0$ for $t>0$,
which ensures that the initial value problem is well-posed. Their paper mainly
discussed the spectrum of the linearized problem, and studied the sign of
principal eigenvalue.

In this paper, we still consider fresh water organisms living in a river
domain ${\Omega}=(l_{1},l_{2})$ and intend to study the spatial dynamics
behaviors of fresh water organisms in advective environment. If we further
assume that these organisms can not enter or leave the domain $\Omega$ and are
affected by the velocity $q$ of currents from left to right, then we study the
following nonlocal Neumann advection-diffusion model:

$\begin{cases}u_{t}=d\int_{\Omega}J(x-y)[u(t,y)-u(t,x)]dy-
qu_{x}+f(t,x,u),&t>0,x\in{\Omega},\\\ u(t,l_{1})=0,&t>0,\\\
u(0,x)=u_{0}(x),&x\in{\Omega}.\end{cases}$ (1.3)

If we further assume that these organisms are affected by the velocity $q$ of
currents from left to right and they can not survive on land $(-\infty,l_{1})$
and in the ocean $(l_{2},+\infty)$. Then, we study the following nonlocal
Dirichlet advection-diffusion model:

$\begin{cases}u_{t}=d[\int_{\Omega}J(x-y)u(t,y)dy-u(t,x)]-qu_{x}+f(t,x,u),&t>0,x\in\Omega,\\\
u(t,l_{1})=0,&t>0,\\\ u(0,x)=u_{0}(x),&x\in\Omega.\end{cases}$ (1.4)

Here $\Omega$ is a bounded open domain, $q>0$ is the advection rate and the
initial function $u_{0}(x)$ satisfies

$\displaystyle u_{0}(x)\in
C^{1}(\bar{\Omega}),u_{0}(l_{1})=0,u_{0}(x)\geq,\not\equiv 0\ \ \mbox{in}\ \
{\Omega}.$ (1.5)

######  Remark 1.1.

  1. (1)

The boundary condition $u(t,l_{1})=0$ for $t>0$ is to ensure the initial value
problem is well-posed. In fact, the equation (1.3) or (1.4) are essentially a
perturbation of a transport equation by an integral operator. When $d\equiv
0$, we find that the equation is a transport equation in a bounded domain,
which need a boundary condition to ensure the well-posedness of solution.

  2. (2)

If these organisms are affected by the velocity $q$ of currents from right to
left, the boundary condition will be rewritten as $u(t,l_{2})=0$ for $t>0$.
Its discussion is similar to the analysis of (1.3) or (1.4).

Assume further that the kernel function $J:\mathbb{R}\to\mathbb{R}^{+}$
satisfies the properties

  1. (J0):

There exist $\mu,M>0$ such that $J(x)<e^{-\mu|x|}$ and
$|J^{\prime}(x)|<e^{-\mu|x|}$ for $|x|>M$.

  1. (J1):

$J$ is symmetric, $\sup J(x)<\infty$.

Meanwhile, assume that, for any given $\tilde{B}>\tilde{b}$, there exists
constant $N>0$ such that for any
$(t,x),(s,y)\in\mathbb{R}^{+}\times\overline{\Omega}$ and
$u\in[\tilde{b},\tilde{B}]$, the function $f(t,x,u)$ satisfies

  1. (F1):

$f,f_{t},f_{x},f_{u}$ are continuous in $t,x,u$; $f,f_{u}$ are Lipschitz
continuous in $u$;

  1. (F2):

$f(t,x,0)=0$; $f(t,x,u)<0,\forall u>N$.

The purpose of this paper is to show the existence and uniqueness of global
solutions, the existence/nonexistence, uniqueness and stability of nontrivial
stationary solutions, the longtime behaviors of solutions, and get the sharp
criteria for persistence or extinction. In addition, the effects of advection
rate on the dynamic behaviors of solutions will be discussed.

We point out that, we adopt a completely new method to get the well-posedness
of global solution of (1.3) and (1.4). Due to the lack of the compactness and
regularity of the nonlocal dispersal operator, the well-posed global solution
of (1.3) and (1.4) will be proved by approximating the solution of a
regularized problem containing a viscosity term, where the upper and lower
solution methods, maximum principle and iterative techniques play an important
role. Moreover, in the process of discussing nontrivial stationary solutions,
it is necessary to consider the spectral theory of integro-differential
operator. In this paper, we aim to solve an open problem posed in [44] under
certain conditions. Specifically, we prove that if $\Omega$ is bounded and
$a(x)\in C(\bar{\Omega})\cap L^{\infty}(\Omega)$ is nonnegative, then the
principal eigenvalue of operator $\mathfrak{L}_{\Omega,J,q}+a$ defined in
(3.2) is increasing in $|q|>0$. The above conclusion can directly deduce the
sharp criteria for persistence or extinction. Meanwhile, the proof methods
about the uniqueness of nontrivial stationary solutions of (1.3) or (1.4) are
different from the usual methods, since all nontrivial stationary solutions of
(1.3) or (1.4) are $0$ at left boundary point $l_{1}$.

The remaining parts of this paper are organized as follows. Section 2 is
devoted to studying the existence and uniqueness of solutions for (1.3) and
(1.4). In Section 3, the existence/nonexistence, uniqueness and stability of
nontrivial stationary solutions are obtained. In Section 4, the criteria for
persistence or extinction are formulated. And in Section 5, the limiting
profiles of solution with respect to advection rate $q$ are considered. The
last section gives a brief discussion.

##  2 The well-posed global solution

For convenience, we set

$Q_{T}=(0,T)\times\Omega,Q_{\infty}=\mathbb{R}^{+}\times\bar{\Omega},\overline{Q_{T}}=[0,T]\times\bar{\Omega}.$

In order to obtain the existence of solutions for nonlocal Neumann problem
(1.3), we must consider the following linear problem:

$\begin{cases}u_{t}=d\int_{\Omega}J(x-y)[u(t,y)-u(t,x)]dy-
qu_{x}+c(t,x)u+g(t,x),&t>0,x\in{\Omega},\\\ u(t,l_{1})=0,&t>0,\\\
u(0,x)=u_{0}(x),&x\in{\Omega}.\end{cases}$ (2.1)

where $c(t,x),g(t,x)\in C^{1}(Q_{\infty})\cap L^{\infty}(Q_{\infty})$.

######  Definition 2.1.

We say that $u(t,x)\in W^{1,2}(\overline{Q_{T}})$ is a local weak solution of
(2.1) if

$\int_{\Omega}\int_{0}^{T}\phi
u_{t}dtdx=\int_{\Omega}\int_{0}^{T}[d\int_{\Omega}J(x-y)[u(t,y)-u]dy-
qu_{x}+cu+g]\phi dtdx$

for all $\phi\in C_{0}^{\infty}(\overline{Q_{T}})$, where

$C_{0}^{\infty}(\overline{Q_{T}}):=\\{\phi\in C^{\infty}(\overline{Q_{T}})|\
D^{k}_{x}\phi(t,l_{2})=0\ \forall t\in(0,T),\forall k\in\mathbb{N}\\}.$

Due to the lack of compactness and regularity of nonlocal operators, many
classical methods cannot be used directly to nonlocal dispersal problems. To
overcome these difficulties, we shall consider the following problem
regularized by the usual viscosity term:

$\begin{cases}u_{t}^{\epsilon}-\epsilon
u^{\epsilon}_{xx}=d\int_{\Omega}J(x-y)[u^{\epsilon}(t,y)-u^{\epsilon}(t,x)]dy-
qu^{\epsilon}_{x}+c(t,x)u^{\epsilon}+g(t,x),&t>0,x\in{\Omega},\\\
u^{\epsilon}(t,l_{1})=\frac{\partial
u^{\epsilon}}{\partial\bf{\overrightarrow{n}}}(t,l_{2})=0,&t>0,\\\
u^{\epsilon}(0,x)=u_{0}(x),&x\in\Omega,\end{cases}$ (2.2)

where $\overrightarrow{n}$ denotes the outward pointing unit normal along
$\partial\Omega$. Our first goal is to show that problem (2.2) has a unique
local nonnegative solution.

######  Lemma 2.2.

For fixed $\epsilon>0$, if $u_{0}(x)$ satisfies (1.5), then there exists $T>0$
such that the regularized problem (2.2) admits a unique nonnegative local
classical solution $u^{\epsilon}(t,x)$ satisfying

$\displaystyle 0\leq\mbox{ess}\inf_{\bar{\Omega}}$ $\displaystyle u_{0}(x)\leq
u^{\epsilon}(t,x)\leq\mbox{ess}\sup_{\bar{\Omega}}u_{0}(x),\ \
\forall(t,x)\in\overline{Q_{T}},$ (2.3)

and

$\displaystyle|u^{\epsilon}_{x}(t,x)|\leq R,\ \
\forall(t,x)\in\overline{Q_{T}},$ (2.4)

where

$R:=\max\\{|\min_{\bar{\Omega}}u_{0}^{\prime}(x)|,|\max_{\bar{\Omega}}u_{0}^{\prime}(x)|,\max_{[0,T]}u^{\epsilon}_{x}(t,l_{1})\\}.$

###### Proof.

Set $X=L^{\infty}(0,T;H^{1}(\Omega))$. For each $v^{\epsilon}\in X$, consider
the linear problem

$\begin{cases}u_{t}^{\epsilon}-\epsilon
u_{xx}^{\epsilon}=d\int_{\Omega}J(x-y)[v^{\epsilon}(t,y)-v^{\epsilon}(t,x)]dy-
qv^{\epsilon}_{x}+c(t,x)v^{\epsilon}+g(t,x),&t\in(0,T),x\in\Omega,\\\
u^{\epsilon}(t,l_{1})=\frac{\partial
u^{\epsilon}}{\partial\bf{\overrightarrow{n}}}(t,l_{2})=0,&t\in(0,T),\\\
u^{\epsilon}(0,x)=u_{0}(x),&x\in\Omega.\end{cases}$ (2.5)

Notice that the right-hand side of the first equation for (2.5) is bounded on
$L^{2}$ since nonlocal operator is bounded on $X$ and $v^{\epsilon}_{x}\in
L^{\infty}(0,T;L^{2}(\Omega))$. Then the standard $L^{p}$ theory implies that
(2.5) admits a unique solution $u^{\epsilon}\in L^{2}(0,T;H^{2}(\Omega))$.
Similarly, choosing $\tilde{v}^{\epsilon}\in X$, and then
$\tilde{u}^{\epsilon}\in L^{2}(0,T;H^{2}(\Omega))$ solves

$\begin{cases}\tilde{u}_{t}^{\epsilon}-\epsilon\tilde{u}_{xx}^{\epsilon}=d\int_{\Omega}J(x-y)[\tilde{v}^{\epsilon}(t,y)-\tilde{v}^{\epsilon}(t,x)]dy-q\tilde{v}^{\epsilon}_{x}+c(t,x)\tilde{v}^{\epsilon}+g(t,x),&t\in(0,T),x\in\Omega,\\\
u^{\epsilon}(t,l_{1})=\frac{\partial
u^{\epsilon}}{\partial\bf{\overrightarrow{n}}}(t,l_{2})=0,&t\in(0,T),\\\
\tilde{u}^{\epsilon}(0,x)=u_{0}(x),&x\in\Omega.\end{cases}$

Hence, for given $z^{\epsilon}:=v^{\epsilon}-\tilde{v}^{\epsilon}$, we find
that $w^{\epsilon}:=u^{\epsilon}-\tilde{u}^{\epsilon}$ satisfies

$\begin{cases}w_{t}^{\epsilon}-\epsilon
w_{xx}^{\epsilon}=d\int_{\Omega}J(x-y)[z^{\epsilon}(t,y)-z^{\epsilon}(t,x)]dy-
qz^{\epsilon}_{x}+c(t,x)z^{\epsilon},&t\in(0,T),x\in\Omega,\\\
w^{\epsilon}(t,l_{1})=\frac{\partial
w^{\epsilon}}{\partial\bf{\overrightarrow{n}}}(t,l_{2})=0,&t\in(0,T),\\\
w^{\epsilon}(0,x)=0,&x\in\Omega.\end{cases}$

Since it follows from [47, Theorem 2, Chapter 5.9.2] that

$\mbox{ess}\sup_{[0,T]}\|w^{\epsilon}\|_{H^{1}(\Omega)}\leq
C(\epsilon)\|w^{\epsilon}\|_{W^{1,2}(0,T;L^{2}(\Omega))},$

we have the following estimation

$\displaystyle\mbox{ess}\sup_{[0,T]}$
$\displaystyle\|w^{\epsilon}\|_{H^{1}(\Omega)}\leq
C(\epsilon)\|w^{\epsilon}\|_{W^{1,2}(0,T;L^{2}(\Omega))}\leq
C(\epsilon)\|w^{\epsilon}\|_{W^{2,2}(0,T;L^{2}(\Omega))}$ $\displaystyle\leq
C(\epsilon)(d\|\int_{\Omega}J(x-y)[z^{\epsilon}(t,y)-z^{\epsilon}(t,x)]dy\|_{L^{2}(0,T;L^{2}(\Omega))}+\|qz^{\epsilon}_{x}+c(t,x)z^{\epsilon}\|_{L^{2}(0,T;L^{2}(\Omega))})$
$\displaystyle\leq
C(\epsilon)(2d\|J\|_{L^{2}(0,T;L^{2}(\Omega))}\|z^{\epsilon}\|_{L^{2}(0,T;L^{2}(\Omega))}+q\|z^{\epsilon}\|_{L^{2}(0,T;H^{1}(\Omega))}+\|c\|_{C^{1}(\overline{Q_{T}})}\|z^{\epsilon}\|_{L^{2}(0,T;L^{2}(\Omega))})$
$\displaystyle\leq C(\epsilon)\|z^{\epsilon}\|_{L^{2}(0,T;H^{1}(\Omega))}\leq
C(\epsilon)T^{1/2}\mbox{ess}\sup_{[0,T]}\|z^{\epsilon}\|_{H^{1}(\Omega)},$

where we use embedding theory and standard $L^{p}$ theory.

As long as $T\in(0,\frac{1}{4C^{2}(\epsilon)}]$, we have
$\|u^{\epsilon}-\tilde{u}^{\epsilon}\|_{L^{\infty}(0,T;H^{1}(\Omega))}\leq\frac{1}{2}\|v^{\epsilon}-\tilde{v}^{\epsilon}\|_{L^{\infty}(0,T;H^{1}(\Omega))}$,
which means that the mapping $A:X\to X$ for $A(v^{\epsilon})=u^{\epsilon}$ is
a contraction mapping. Based on Banach’s Fixed Point Theorem, mapping $A$ has
a unique fixed point, and so (2.2) has a unique local-in-time solution
$u^{\epsilon}\in L^{\infty}(0,T;H^{1}(\Omega))$. By embedding theory and
standard Schauder theory, we can obtain that $u^{\epsilon}$ also is a local
classical solution.

We next examine the inequalities (2.3). Let
$M=\mbox{ess}\sup_{\bar{\Omega}}u_{0}(x)>0$. The function
$w(t,x):=u^{\epsilon}(t,x)-M$ satisfies

$\begin{cases}w_{t}=\epsilon w_{xx}+d\int_{\Omega}J(x-y)[w(t,y)-w(t,x)]dy-
qw_{x}+c(t,x)(w+M)+g(t,x),&t\in(0,T),x\in\Omega,\\\
w(t,l_{1})=-M<0,\frac{\partial
w}{\partial\bf{\overrightarrow{n}}}(t,l_{2})=0,&t\in(0,T),\\\
w(0,x)=u_{0}(x)-M\leq 0,&x\in\Omega.\end{cases}$

Claim: The inequality $w(0,x)\leq 0$ implies $w(t,x)\leq 0$ for all
$(t,x)\in\overline{Q_{T}}$. Define function $w^{1}=e^{-Kt}w$ and suppose that
there is $(t_{*},x_{*})\in\overline{Q_{T}}$ such that
$w^{1}(t_{*},x_{*})=\sup_{\overline{Q_{T}}}w^{1}(t,x)>0$. Obviously,
$t_{*}\in(0,T]$ and $x_{*}\in(l_{1},l_{2}]$, and then
$w^{1}_{t}(t_{*},x_{*})\geq 0$. Meanwhile, we can choose large enough constant
$K>0$ satisfying

$(c(t_{*},x_{*})-K)w^{1}(t_{*},x_{*})+(c(t_{*},x_{*})M+g(t_{*},x_{*}))e^{-Kt_{*}}<0\
\mbox{on}\ \overline{Q_{T}},$

since $c(t,x),g(t,x)\in C^{1}(\overline{Q_{T}})$.

If $x_{*}\in\Omega$, then it follows from $w^{1}_{xx}(t_{*},x_{*})\leq
0,w^{1}_{x}(t_{*},x_{*})=0$ and $w^{1}(t_{*},x)\leq w^{1}(t_{*},x_{*})$ on
$\Omega$ that

$\displaystyle 0\leq w^{1}_{t}(t_{*},x_{*})=$ $\displaystyle
e^{-Kt_{*}}w_{t}(t_{*},x_{*})-Ke^{-Kt_{*}}w(t_{*},x_{*})$
$\displaystyle=\epsilon
w^{1}_{xx}(t_{*},x_{*})+d\int_{\Omega}J(x^{*}-y)[w^{1}(t_{*},y)-w^{1}(t_{*},x_{*})]dy-
qw^{1}_{x}(t_{*},x_{*})$
$\displaystyle\quad+(c(t_{*},x_{*})-K)w^{1}(t_{*},x_{*})+(c(t_{*},x_{*})M+g(t_{*},x_{*}))e^{-Kt_{*}}$
$\displaystyle\leq(c(t_{*},x_{*})-K)w^{1}(t_{*},x_{*})+(c(t_{*},x_{*})M+g(t_{*},x_{*}))e^{-Kt_{*}}<0,$

which is impossible. If $x_{*}=l_{2}$, the boundary condition $\frac{\partial
w}{\partial\bf{\overrightarrow{n}}}|_{x=l_{2}}=0$ implies
$w^{1}_{x}(t_{*},x_{*})=0$. We slightly modify the proof by setting
$w^{1}(t,x_{n})=w^{1}(t,x_{*})-\dfrac{1}{n}$. By [48, Lemma 3.1], the sequence
$\\{w^{1}(t,x_{n})\\}$ has the following properties:

$\lim\limits_{n\to\infty}w^{1}_{x}(t,x_{n})=0;\
\limsup\limits_{n\to\infty}w^{1}_{xx}(t,x_{n})\leq 0;\
\limsup\limits_{n\to\infty}\int_{\Omega}J(x_{n}-y)[w^{1}(t,y)-w^{1}(t,x_{n})]dy\leq
0.$

Thus, letting $t=t_{*}$, as $n\to\infty$, we have

$\displaystyle 0\leq
w^{1}_{t}(t_{*},x_{*})\leq(c(t_{*},x_{*})-K)w^{1}(t_{*},x_{*})+(c(t_{*},x_{*})M+g(t_{*},x_{*}))e^{-Kt_{*}}<0,$

which is impossible. And the claim is true. Thus $u^{\epsilon}(t,x)\leq M$ on
$\overline{Q_{T}}$, and the proof of the inequality
$0\leq\mbox{ess}\inf_{\bar{\Omega}}u_{0}(x)\leq u^{\epsilon}(t,x)$ is
analogous.

Finally, we verify the inequality (2.4). Denote
$z(t,x)=u^{\epsilon}_{x}(t,x)$. By assumption ${\bf{(J1)}}$, we know that
$z(t,x)$ satisfies

$\begin{cases}z_{t}=\epsilon z_{xx}+d\int_{\Omega}J(x-y)[z(t,y)-z(t,x)]dy-
qz_{x}+cz+c_{x}u^{\epsilon}+g_{x}-m(t,x),&t\in(0,T),x\in\Omega,\\\
z(t,l_{1})=u^{\epsilon}_{x}(t,l_{1})\geq 0,z(t,l_{2})=0,&t\in(0,T),\\\
z(0,x)=(u_{0})^{\prime}(x),&x\in\Omega,\end{cases}$

where

$\displaystyle
m(t,x):=du^{\epsilon}(t,l_{2})J(x-l_{2})-du^{\epsilon}(t,l_{1})J(x-l_{1})+d\int_{\Omega}J_{x}(x-y)dyu^{\epsilon}$
(2.6)

is uniformly bounded. Since $u^{\epsilon}$ is a uniformly bounded classical
solution, $c_{x}u^{\epsilon}+g_{x}-m(t,x)$ is a regular function, and then

$|c_{x}u^{\epsilon}+g_{x}-m(t,x)|\leq\|c\|_{C^{1}}\|u_{0}\|_{\infty}+\|g\|_{C^{1}}+\|m\|_{\infty}:=R_{1}.$

Let $v(t,x):=(R-z(t,x))e^{-Kt}$ for some large enough $K>0$. Clearly,
$v(0,x)=R-z(0,x)\geq 0$ and $v(t,x)\geq 0$ on $\partial\Omega$. We shall show
$v(t,x)\geq 0$ in $\overline{Q_{T}}$. Conversely, if there is
$(t_{*},x_{*})\in\overline{Q_{T}}$ such that
$v(t_{*},x_{*})=\inf_{\overline{Q_{T}}}v(t,x)<0$. The choice of $R$ deduces
$t_{*}\in(0,T]$ and $v_{t}(t_{*},x_{*})\leq 0$. The boundary condition means
$x_{*}\in(l_{1},l_{2})$, $v_{x}(t_{*},x_{*})=0$ and $v_{xx}(t_{*},x_{*})\geq
0$. In view of the assumption ${\bf{(J1)}}$, we know that

$d\int_{\Omega}J(x_{*}-y)[v(t_{*},y)-v(t_{*},x_{*})]dy\geq 0.$

Then $v(t_{*},x_{*})$ satisfies

$\displaystyle 0\geq v_{t}(t_{*},x_{*})=$ $\displaystyle-
Kv(t_{*},x_{*})+\epsilon
v_{xx}(t_{*},x_{*})+d\int_{\Omega}J(x_{*}-y)[v(t_{*},y)-v(t_{*},x_{*})]dy-
qv_{x}(t_{*},x_{*})$
$\displaystyle+c(t_{*},x_{*})v(t_{*},x_{*})+[c_{x}(t_{*},x_{*})u^{\epsilon}(t_{*},x_{*})+g_{x}(t_{*},x_{*})-m(t_{*},x_{*})-Rc(t_{*},x_{*})]e^{-Kt_{*}}$
$\displaystyle\geq$
$\displaystyle(c(t_{*},x_{*})-K)v(t_{*},x_{*})-(R_{1}+R\|c\|_{C^{1}})e^{-Kt_{*}}$
$\displaystyle\geq$
$\displaystyle(\|c\|_{C^{1}}-K)v(t_{*},x_{*})-(R_{1}+R\|c\|_{C^{1}})e^{-Kt_{*}}>0,$

provided that $K>0$ is a large enough constant such that

$(\|c\|_{C^{1}}-K)v(t_{*},x_{*})-(R_{1}+R\|c\|_{C^{1}})e^{-Kt_{*}}>0.$

This is a contradiction, and so, $z(t,x)\leq R$ on $\overline{Q_{T}}$.
Similarly, we also can prove $z(t,x)\geq-R$ on $\overline{Q_{T}}$. The proof
is completed. $\Box$

In the following, we prove that the solution $u^{\epsilon}(t,x)$ of
regularized problem (2.2) converges to a weak solution of problem (2.1) as
$\epsilon\to 0$. We will apply Aubin-Lions-Simon compactness theorem
(Proposition 2.3) and Lebesgue dominated theorem to complete this proof.

######  Proposition 2.3 ([49], Theorem 5).

For $1\leq p_{1},p_{2}\leq\infty$, and $T>0$, assume that Banach spaces
$X,B,Y$ satisfy $X\hookrightarrow\hookrightarrow B\hookrightarrow Y$. If $A$
is bounded in $W^{1,p_{1}}(0,T;Y)$ and $L^{p_{2}}(0,T;X)$, then $A$ is
relatively compact in $L^{p_{2}}(0,T;B)$ if $1\leq p_{2}<\infty$ and in
$C(0,T;B)$ if $p_{2}=\infty$, where $X\hookrightarrow\hookrightarrow B$ means
that $X$ is compactly embedded in $B$.

######  Theorem 2.4.

Assume that $\bf{(J)(J1)}$ holds. Then (2.1) admits a local weak solution
$u(t,x)\in L^{\infty}(\overline{Q_{T}})$, where $T$ is given by Lemma 2.2.

###### Proof.

First of all, we use Proposition 2.3 to show that
$\\{u^{\epsilon}(t,x)|\epsilon\in(0,1]\\}$ is relatively compact in
$C(0,T;L^{2}(\Omega))$. Choosing $p_{1}=p_{2}=\infty$ and
$X=H^{1}(\Omega),B=L^{2}(\Omega),Y=H^{-1}(\Omega)=(H^{1}_{0}(\Omega))^{*}$,
where we define
$\|u\|_{H^{-1}(\Omega)}=\sup\\{\int_{\Omega}(DuD\varphi+u\varphi)dx|\varphi\in
C^{\infty}_{0}(\Omega)\\}$. Obviously,
$H^{1}(\Omega)\hookrightarrow\hookrightarrow L^{2}(\Omega)\hookrightarrow
H^{-1}(\Omega)$. Then for any $\varphi\in C^{\infty}_{0}(\Omega)$,

$\displaystyle|\int_{\Omega}(Du^{\epsilon}D\varphi+u^{\epsilon}\varphi)dx|$
$\displaystyle\leq|\int_{\Omega}Du^{\epsilon}D\varphi
dx|+|\int_{\Omega}u^{\epsilon}\varphi
dx|\leq|\int_{\Omega}u^{\epsilon}D^{2}\varphi
dx|+|\int_{\Omega}u^{\epsilon}\varphi dx|$
$\displaystyle\leq\|u_{0}\|_{\infty}(\int_{\Omega}|D^{2}\varphi|+|\varphi|dx)\leq
2\|u_{0}\|_{\infty}\|\varphi\|_{C_{0}^{\infty}}|\Omega|<\infty.$

Notice that

$\displaystyle\int_{\Omega}|u^{\epsilon}_{t}D^{2}\varphi|dx$
$\displaystyle\leq\int_{\Omega}\left|[\epsilon
u_{xx}^{\epsilon}+d\int_{\Omega}J(x-y)[u^{\epsilon}(t,y)-u^{\epsilon}(t,x)]dy-
qu^{\epsilon}_{x}+c(t,x)u^{\epsilon}+g(t,x)]D^{2}\varphi\right|dx$
$\displaystyle\leq\int_{\Omega}\left[\epsilon|u^{\epsilon}D^{4}\varphi|+2d\|u_{0}\|_{\infty}\|\varphi\|_{C_{0}^{\infty}}+|qu^{\epsilon}D^{3}\varphi|+\|c\|_{\infty}|u^{\epsilon}D^{2}\varphi|\right]dx+\int_{\Omega}\|g\|_{\infty}|D^{2}\varphi|dx$
$\displaystyle\leq\|\varphi\|_{C_{0}^{\infty}}|\Omega|\|u_{0}\|_{\infty}(1+2d+|q|+\|c\|_{\infty})+\|\varphi\|_{C_{0}^{\infty}}|\Omega|\|g\|_{\infty}<\infty.$

Similarly,
$\int_{\Omega}|u^{\epsilon}_{t}\varphi|dx\leq\|\varphi\|_{C_{0}^{\infty}}|\Omega|\|u_{0}\|_{\infty}(1+2d+|q|+\|c\|_{\infty})+\|\varphi\|_{C_{0}^{\infty}}|\Omega|\|g\|_{\infty}<\infty$.
Thus, we have

$\displaystyle|\int_{\Omega}(Du^{\epsilon}_{t}D\varphi+u^{\epsilon}_{t}\varphi)dx|\leq\int_{\Omega}|u^{\epsilon}_{t}D^{2}\varphi|dx+\int_{\Omega}|u^{\epsilon}_{t}\varphi|dx<\infty.$

The above estimates have applied inequalities (2.3) and integrating by part
formula. It means that
$\|u^{\epsilon},u^{\epsilon}_{t}\|_{H^{-1}(\Omega)}<\infty$, namely, there
exists a constant $C>0$ independent of $\epsilon$ such that

$\|u^{\epsilon}\|_{W^{1,\infty}(0,T;H^{-1}(\Omega))}=\mbox{ess}\
sup_{[0,T]}\|u^{\epsilon}\|_{H^{-1}(\Omega)}+\mbox{ess}\
sup_{[0,T]}\|u^{\epsilon}_{t}\|_{H^{-1}(\Omega)}\leq C<\infty.$

And so, $\\{u^{\epsilon}(t,x)|\epsilon\in(0,1]\\}$ is bounded in
$W^{1,\infty}(0,T;H^{-1}(\Omega))$.

Obviously,
$|\int_{\Omega}(u^{\epsilon})^{2}dx|\leq\|u_{0}^{\epsilon}\|_{\infty}^{2}|\Omega|$
and $|\int_{\Omega}(u^{\epsilon})_{x}^{2}dx|\leq R^{2}|\Omega|$ by Lemma 2.2,
that is to say, $\\{u^{\epsilon}(t,x)|\epsilon\in(0,1]\\}$ is bounded in
$L^{\infty}(0,T;H^{1}(\Omega))$. Hence, Proposition 2.3 ensures that
$\\{u^{\epsilon}(t,x)|\epsilon\in(0,1]\\}$ is relatively compact in
$C(0,T;L^{2}(\Omega))$, and then there exist
$\\{\epsilon_{n}\\}_{n\in\mathbb{N}}\subset(0,1]$ and function $u$ such that
$u^{\epsilon_{n}}$ converges to $u$ in $C(0,T;L^{2}(\Omega))$ as
$\epsilon_{n}\to 0$. Combining the inequalities (2.3), we have $u\in
L^{\infty}(\overline{Q_{T}})$.

Now, we show that $u(t,x)\in L^{\infty}(\overline{Q_{T}})$ is a local weak
solution of (2.1). Multiplying the first equation of (2.2) by $\phi\in
C_{0}^{\infty}(\overline{Q_{T}})$, where

$C_{0}^{\infty}(\overline{Q_{T}}):=\\{\phi\in
C^{\infty}(\overline{Q_{T}})|D^{k}_{x}\phi(t,l_{2})=0\ \forall
t\in(0,T),\forall k\in\mathbb{N}\\},$

and integrating over $[0,T]\times\Omega$, then using the integrating by part
formula and the condition $\bf{(J1)}$, we deduce

$\displaystyle\int_{\Omega}$
$\displaystyle\phi(T,x)u^{\epsilon_{n}}(T,x)dx-\int_{\Omega}\phi(0,x)u^{\epsilon_{n}}_{0}(x)dx-\int_{\Omega}\int_{0}^{T}\phi_{t}u^{\epsilon_{n}}dtdx$
$\displaystyle=$ $\displaystyle
d\int_{\Omega}\int_{0}^{T}\int_{\Omega}J(x-y)[\phi(t,y)-\phi(t,x)]dyu^{\epsilon_{n}}dtdx+q\int_{\Omega}\int_{0}^{T}\phi_{x}u^{\epsilon_{n}}dtdx$
$\displaystyle+\epsilon_{n}\int_{\Omega}\int_{0}^{T}\phi_{xx}u^{\epsilon_{n}}dtdx+\int_{\Omega}\int_{0}^{T}(c\phi
u^{\epsilon_{n}}+g\phi)dtdx+\epsilon_{n}\int_{0}^{T}u^{\epsilon_{n}}_{x}(t,l_{1})\phi(t,l_{1})dt.$

As $\epsilon_{n}\to 0$, applying the Lebesgue dominated theorem, we obtain

$\displaystyle\int_{\Omega}$
$\displaystyle\phi(T,x)u(T,x)dx-\int_{\Omega}\phi(0,x)u_{0}(x)dx-\int_{\Omega}\int_{0}^{T}\phi_{t}udtdx$
$\displaystyle=d\int_{\Omega}\int_{0}^{T}\int_{\Omega}J(x-y)[\phi(t,y)-\phi(t,x)]dyu(t,x)dtdx+\int_{\Omega}\int_{0}^{T}(q\phi_{x}u+c\phi
u+g\phi)dtdx.$

Again using the integrating by part formula, $u$ satisfies

$\displaystyle\int_{\Omega}\int_{0}^{T}\phi
u_{t}dtdx=\int_{\Omega}\int_{0}^{T}[d\int_{\Omega}J(x-y)[u(t,y)-u]dy-
qu_{x}+cu+g]\phi dtdx-\int_{0}^{T}u(t,l_{1})\phi(t,l_{1})dt,$

which verifies that $u$ is a weak solution of (2.1) since the boundary
condition $u(t,l_{1})=0$. The proof is completed. $\Box$

######  Remark 2.5.

As shown in the proof of [50, Lemma 2,2], since $f(t,x,u)$ satisfies (F1)(F2),
we know that $u\in C^{1}(\overline{Q_{T}})$ and $u$ is also a classical
solution of (2.1).

######  Lemma 2.6 (Maximum principle).

For any given $u_{0}(x)\geq 0$ and $T>0$, assume that, for some $c(t,x)\in
L^{\infty}(\overline{Q_{T}})$, function $u(t,x)\in C^{1}(\overline{Q_{T}})$
satisfies

$\begin{cases}u_{t}\geq d\int_{\Omega}J(x-y)[u(t,y)-u(t,x)]dy-
qu_{x}+c(t,x)u,&t\in(0,T),x\in{\Omega},\\\ u(t,l_{1})\geq 0,&t\in(0,T),\\\
u(0,x)=u_{0}(x)\geq 0,&x\in{\Omega}.\end{cases}$ (2.7)

Then $u(t,x)\geq 0$ in $\overline{Q_{T}}$. Moreover, $u(t,x)>0$ in
$(0,T]\times(l_{1},l_{2}]$ if $u_{0}(x)\not\equiv 0$ in $\bar{\Omega}$.

###### Proof.

Choose a negative constant $k$ such that $k+c(t,x)<0$ in $\overline{Q_{T}}$.
Then $w(t,x):=e^{kt}u(t,x)$ satisfies

$\displaystyle w_{t}\geq d\int_{\Omega}J(x-y)[w(t,y)-w(t,x)]dy-
qw_{x}+(c(t,x)+k)w.$ (2.8)

Suppose that there exists $(t_{0},x_{0})\in\overline{Q_{T}}$ such that
$w_{\inf}:=w(t_{0},x_{0})=\inf_{\overline{Q_{T}}}w(t,x)<0$. Similar to the
proof of Lemma 2.2, we deduce

$\displaystyle 0\geq w_{t}(t_{0},x_{0})$ $\displaystyle\geq
d\int_{\Omega}J(x_{0}-y)[w(t_{0},y)-w(t_{0},x_{0})]dy-
qw_{x}(t_{0},x_{0})+(c(t_{0},x_{0})+k)w(t_{0},x_{0})$
$\displaystyle\geq(c(t_{0},x_{0})+k)w(t_{0},x_{0})>0,$

which is a contradiction. Thus, $w(t,x)\geq 0$ in $\overline{Q_{T}}$, namely,
$u(t,x)\geq 0$ in $\overline{Q_{T}}$.

Now, we are still in a position to prove that $u(t,x)>0$ in
$(0,T]\times(l_{1},l_{2}]$ if $u_{0}(x)\not\equiv 0$ in ${\Omega}$. It is
equivalent to proving $w(t,x)>0$ in $(0,T]\times(l_{1},l_{2}]$. Suppose that
there is $(t^{*},x^{*})\in(0,T]\times(l_{1},l_{2}]$ such that
$w(t^{*},x^{*})=0$. We observe that $w(t^{*},x)\equiv 0$ for any
$x\in\bar{\Omega}$. Otherwise, we can find a sequence
$\\{x_{n}\\}_{n=1}^{\infty}\subset{\Omega}$ with $x_{n}\to x^{*}$ such that
$w(t^{*},x_{n})>0$ and $w(t^{*},x_{n})\to w(t^{*},x^{*})$ as $n\to\infty$.
Noting that $w_{t}(t^{*},x^{*})\leq 0$ and $w_{x}(t^{*},x^{*})\leq 0$, we
obtain

$0\geq w_{t}(t^{*},x^{*})\geq
d\int_{\Omega}J(x^{*}-y)[w(t^{*},y)-w(t^{*},x^{*})]dy-
qw_{x}(t^{*},x^{*})+(c(t^{*},x^{*})+k)w(t^{*},x^{*}),$

which implies $\int_{\Omega}J(x^{*}-y)w(t^{*},y)dy=0$. In view of $J(0)>0$ and
$w(t^{*},y)\geq 0$ in $\bar{\Omega}$, it is easy to get $w(t^{*},y)=0$ for all
$y$ near $x^{*}$, which is a contradiction with $w(t^{*},x_{n})>0$. Define
$z(s)=w(t+s,x+qs)$. By a simple calculation, we have

$\displaystyle z^{\prime}(s)$ $\displaystyle=w_{t}(t+s,x+qs)+qw_{x}(t+s,x+qs)$
$\displaystyle\geq
d\int_{\Omega}J(x+qs-y)[w(t+s,y)-w(t+s,x+qs)]dy+[c(t+s,x+qs)+k]w(t+s,x+qs)$
$\displaystyle\geq\inf_{{Q_{\infty}}}[k+c(t,x)-d]z(s)\triangleq\theta z(s).$

And so, $[e^{-\theta s}z(s)]^{\prime}\geq 0$, namely, for any $t\in[0,t^{*}]$,
it follows that

$e^{-\theta(t^{*}-t)}z(t^{*}-t)=e^{-\theta(t^{*}-t)}w(t^{*},x+q(t^{*}-t))\equiv
0\geq z(0)=w(t,x)\geq 0.$

Thus, $w(t,x)\equiv 0$ in $[0,t^{*}]\times\bar{\Omega}$, and it contradicts
with $u_{0}(x)\not\equiv 0$ in $\bar{\Omega}$. The proof is completed. $\Box$

######  Theorem 2.7.

Assume that $\bf{(J)(J1)}$ holds. Then (2.1) admits a unique local classical
solution $u(t,x)\in C^{1}(\overline{Q_{T}})$, where $T$ is given by Lemma 2.2.

###### Proof.

By Remark 2.5, we know that (2.1) admits a local classical solution $u(t,x)$.
Here, we only prove the uniqueness of classical solution $u$. Suppose that for
any given $u_{0}(x)$, (2.1) has two different solution $u_{1}$ and $u_{2}$.
Then $w:=u_{1}-u_{2}$ satisfies

$\begin{cases}w_{t}=d\int_{\Omega}J(x-y)[w(t,y)-w(t,x)]dy-
qw_{x}+c(t,x)w,&t\in(0,T),x\in{\Omega},\\\ w(t,l_{1})=0,&t\in(0,T),\\\
w(0,x)=0,&x\in{\Omega}.\end{cases}$

By Lemma 2.6, we have $w\geq 0$ on $\overline{Q_{T}}$, and $w\leq 0$ on
$\overline{Q_{T}}$. And so $u_{1}-u_{2}=0$. It is a contradiction with our
assumption. The uniqueness is completed. $\Box$

In the followings, we shall apply the upper and lower solution methods to get
the existence of solutions of (1.3), so we define the upper and lower
solutions as follows:

######  Definition 2.8.

The function $\bar{u}(t,x)\in C^{1}(\overline{Q_{T}})$ is called the upper
solution of (1.3) if

$\begin{cases}\bar{u}_{t}\geq
d\int_{\Omega}J(x-y)[\bar{u}(t,y)-\bar{u}(t,x)]dy-q\bar{u}_{x}+f(t,x,\bar{u}),&t>0,x\in{\Omega},\\\
\bar{u}(t,l_{1})\geq 0,&t\in(0,T),\\\ \bar{u}(0,x)\geq
u_{0}(x),&x\in{\Omega}.\end{cases}$

Moreover, we can also define a lower solution $\underline{u}(t,x)$ by
reversing all the inequalities in the above equation.

Combining with Lemma 2.6, naturally, it gives a ordering of upper solution and
lower solution:

######  Lemma 2.9.

(Comparison principle ) If $\bar{u}(t,x),\underline{u}(t,x)$ are respectively
an upper solution and a lower solution of (1.3) and $f(t,x,u)$ satisfies
${\bf{(F1)(F2)}}$ with
$\tilde{b}=\min_{\overline{Q_{T}}}\min\\{\bar{u},\underline{u}\\}$ and
$\tilde{B}=\max_{\overline{Q_{T}}}\max\\{\bar{u},\underline{u}\\}$, then
$\bar{u}\geq\underline{u}$ in $Q_{T}$.

According to the assumption ${\bf{(F1)}}$, we choose a large constant $K$
satisfying

$\displaystyle|f(t,x,u_{1})-f(t,x,u_{2})|\leq K|u_{1}-u_{2}|.$ (2.9)

For given upper and lower solutions $\bar{u}(t,x),\underline{u}(t,x)$, by
using $\underline{u}^{(0)}=\underline{u}$ and $\bar{u}^{(0)}=\bar{u}$ as the
initial iterations, we can construct sequences
$\\{\underline{u}^{(m)}\\}^{\infty}_{m=1}$ and
$\\{\bar{u}^{(0)}\\}^{\infty}_{m=1}$ such that the following equation

$\begin{cases}\bar{u}^{(m)}_{t}-dL\bar{u}^{(m)}+q\bar{u}^{(m)}_{x}+K\bar{u}^{(m)}=K\bar{u}^{(m-1)}+f(t,x,\bar{u}^{(m-1)}),&t\in(0,T_{0}),x\in{\Omega},\\\
\underline{u}^{(m)}_{t}-dL\underline{u}^{(m)}+q\underline{u}^{(m)}_{x}+K\underline{u}^{(m)}=K\underline{u}^{(m-1)}+f(t,x,\underline{u}^{(m-1)}),&t\in(0,T_{0}),x\in{\Omega},\\\
\bar{u}^{(m)}(t,l_{1})=\underline{u}^{(m)}(t,l_{1})=0,&t\in(0,T_{0}),\\\
\bar{u}^{(m)}(0,x)=\underline{u}^{(m)}(0,x)=u_{0}(x),&x\in{\Omega}\end{cases}$

holds, where $Lu=\int_{\Omega}J(x-y)[u(t,y)-u(t,x)]dy$. Based on the
assumption ${\bf{(F1)}}$ and Theorem 2.7, it follows that the limits of
sequences $\\{\underline{u}^{(m)}\\}^{\infty}_{m=1}$ and
$\\{\bar{u}^{(m)}\\}^{\infty}_{m=1}$ exist and are unique in
$\overline{Q_{T}}$. Then we can apply a similar argument as in [52, Theorem
3.6], combining with monotone dynamical system methods and Maximum principle
(Lemma 2.6), to get the theorem as follows:

######  Theorem 2.10.

Assume that $\bf{(J),(J1),(F1)}$ and ${\bf{(F2)}}$ hold. For any given
$u_{0}(x)$ satisfying (1.5), if (1.3) has a bounded upper solution
$\bar{u}(t,x)$ and a bounded lower solution $\underline{u}(t,x)$, then there
exists $T>0$ such that (1.3) admits a unique local classical solution
$u(t,x)\in C^{1}(\overline{Q_{T}})$.

We extend the local solution obtained in Theorem 2.10 to the maximum time. To
do so, let $[0,T_{0})$ be the maximum existence interval and $u(t,x)$ be the
solution of (1.3) defined on $[0,T_{0})$.

######  Theorem 2.11.

Assume that $\bf{(J),(J1),(F1)}$ and ${\bf{(F2)}}$ hold. For any given
$u_{0}(x)$ satisfying (1.5), problem (1.3) admits a unique classical solution
$u(t,x)\in C^{1}([0,\infty)\times\bar{\Omega})$. Moreover,

$\displaystyle 0<u(t,x)$
$\displaystyle\leq\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\},\ \
\forall(t,x)\in{Q_{\infty}};$ (2.10) $\displaystyle M_{1}\leq u_{x}(t,x)\leq
M_{2},\ \ \forall(t,x)\in{Q_{\infty}};$ (2.11)

where

$M_{1}=\min\\{\min_{\overline{\Omega}}u_{0}(x),\min_{\overline{\Omega}}u_{0}^{\prime}(x)\\},M_{2}=\max\\{\max_{\overline{\Omega}}u_{0}(x),\max_{\overline{\Omega}}u_{0}^{\prime}(x)\\},$

and $M_{1}$ may be a negative constant and $M_{2}$ is a positive constant.

###### Proof.

Firstly, we shall show $T_{0}=\infty$. Obviously, $\underline{u_{1}}=0$ and
$\bar{u}_{1}=\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\}$ are respectively the
lower and upper solutions, so (1.3) admits the unique solution $u(t,x)$ and
(2.10) holds on $(0,T_{0})\times\bar{\Omega}$ by Lemma 2.6.

Assume $T_{0}<\infty$. Then there is a sequence $\\{T_{n}\\}\subset(0,T_{0})$
such that $T_{n}\to T_{0}$ as $n\to\infty$ and
$\|u\|_{C^{1}(\overline{Q_{T_{n}}})}\leq\bar{u}_{1}$. Since $\bar{u}_{1}$ is
independent of $T_{n}$ and $T_{n}\to T_{0}<\infty$, we have also
$\|u\|_{C^{1}(\overline{Q_{T_{0}}})}\leq\bar{u}_{1}$. Furthermore, the
solution of (1.3) exists on $[0,T_{0}]$. Regarding $T_{n}$ and $u(T_{n},x)$ as
initial time and initial value. Similar to the proof of Theorem 2.10, we can
find a constant $0<t_{0}\ll 1$ such that (1.3) has a unique solution $u_{n}$
in $[T_{n},T_{n}+t_{0}]$. By the uniqueness, $u_{n}=u$ in $T_{n}\leq
t<\min\\{T_{n}+t_{0},T_{0}\\}$. it allows the solution to be extended beyond
$T_{0}$, contradicting with the maximum existence interval. And so, (2.10)
holds on $Q_{\infty}$.

We next shall prove (2.11). Note that ${\bf{(F1)}}$ implies that there exists
a constant $M>0$ such that

$|f_{x}(t,x,u)|,|f_{u}(t,x,u)|\leq M,$

and

$\left|\frac{\partial f(t,x,u)}{\partial
x}\right|=|f_{x}(t,x,u)+f_{u}(t,x,u)u_{x}|\leq M(1+|u_{x}|).$

Denote $z(t,x)=(M_{1}-u_{x})e^{-N_{1}t}$ for some positive constant $N_{1}$
and $z(0,x)=M_{1}-u_{0}^{\prime}(x)\leq 0$. We show $z(t,x)\leq 0$ in
$Q_{\infty}$ by contradiction. Suppose that we can find
$(t_{*},x_{*})\in{Q_{\infty}}$ such that
$z(t_{*},x_{*})=\sup_{{Q_{\infty}}}z(t,x)>0$. The choice of $M_{1}$ deduces
$t_{*}>0$, and the bounbary condition means $x_{*}\in(l_{1},l_{2}]$,
$z_{x}(t_{*},x_{*})\geq 0$ and $z_{t}(t_{*},x_{*})=0$. In view of the
assumption ${\bf{(J1)}}$, we know that $z(t_{*},x_{*})$ satisfies

$\displaystyle 0=z_{t}(t_{*},x_{*})=$ $\displaystyle
d\int_{\Omega}J(x_{*}-y)[z(t_{*},y)-z(t_{*},x_{*})]dy-
qz_{x}(t_{*},x_{*})+(f_{u}(t_{*},x_{*},u(t_{*},x_{*}))-N_{1})z(t_{*},x_{*})$
$\displaystyle-[f_{x}(t_{*},x_{*},u(t_{*},x_{*}))+f_{u}(t_{*},x_{*},u(t_{*},x_{*}))M_{1}+m(t_{*},x_{*})]e^{-N_{1}t_{*}}$
$\displaystyle\leq$
$\displaystyle(f_{u}(t_{*},x_{*},u(t_{*},x_{*}))-N_{1})z(t_{*},x_{*})+(|f_{x}(t,x,u)|+|f_{u}(t,x,u)|M_{1}+|m(t_{*},x_{*})|)e^{-N_{1}t_{*}}$
$\displaystyle\leq$
$\displaystyle(M-N_{1})z(t_{*},x_{*})+(M+MM_{1}+\|m(t,x)\|_{\infty})e^{-N_{1}t_{*}}<0,$

provided that $N_{1}$ is large enough satisfying

$(M-N_{1})z(t_{*},x_{*})+(M+MM_{1}+\|m(t,x)\|_{\infty})e^{-N_{1}t_{*}}<0,$

where $m(t,x)$ is given by (2.6). This is a contradiction, and then
$z(t,x)=(M_{1}-u_{x})e^{-N_{1}t}\leq 0$ on $Q_{\infty}$. That is to say,

$M_{1}\leq u_{x}(t,x),\ \ \forall(t,x)\in{Q_{\infty}}$

Similarly, we can define $w(t,x)=(M_{2}-u_{x})e^{-N_{2}t}$ for some large
enough positive constant $N_{2}$ and $z(0,x)=M_{2}-u_{0}^{\prime}(x)\geq 0$ to
obtain $w(t,x)\geq 0$ on $Q_{\infty}$. So,

$M_{2}\geq u_{x}(t,x),\ \ \forall(t,x)\in{Q_{\infty}}.$

The proof is completed. $\Box$

And we can use the same methods to consider the nonlocal Dirichlet problem
(1.4). On the one hand, we can establish the maximum principle for (1.4):

######  Lemma 2.12 (Maximum principle).

For any given $u_{0}(x)\geq 0$ and $T>0$, assume that, for some $c(t,x)\in
L^{\infty}(\overline{Q_{T}})$, function $u(t,x)\in C^{1}(\overline{Q_{T}})$
satisfies

$\begin{cases}u_{t}\geq
d[\int_{\Omega}J(x-y)u(t,y)dy-u(t,x)]-qu_{x}+c(t,x)u,&t\in(0,T),x\in\Omega,\\\
u(t,l_{1})\geq 0,&t\in(0,T),\\\ u(0,x)=u_{0}(x)\geq 0,&x\in\Omega.\end{cases}$

Then $u(t,x)\geq 0$ in $\overline{Q_{T}}$. Moreover, $u(t,x)>0$ in
$(0,T]\times(l_{1},l_{2}]$ if $u_{0}(x)\not\equiv 0$ in $\bar{\Omega}$.

On the other hand, the well-posedness of solutions of (1.4) is obtained as
follows:

######  Theorem 2.13.

Assume that $\bf{(J),(J1),(F1)}$ and ${\bf{(F2)}}$ hold. For any given
$u_{0}(x)$ satisfying (1.5), (1.4) admits a unique classical solution
$u(t,x)\in C^{1}([0,\infty)\times\bar{\Omega})$ satisfying (2.10) and (2.11).

##  3 Nontrivial stationary solution

In this section, we focus on the specific reaction term. Assume further that
$f(t,x,u)=uh(x,u)$ satisfies the following conditions:

  1. (1)

Function $h(x,u)\in C^{1}(\bar{\Omega}\times\mathbb{R}^{+})$ is decreasing in
$u$ on $\mathbb{R}^{+}$;

  2. (2)

Function $h(x,0)$ may be sign changed.

Then (1.3) and (1.4) become a nonlocal reaction-diffusion-advection Fisher-KPP
equation. For given initial value $u_{0}(x)$, it follows that (1.3) (or (1.4))
admits a unique solution $u(t,x;u_{0})$ satisfying $u(t,x;u_{0})>0$ for all
$t>0$.

Moreover, if the nontrivial stationary $U(x)$ of (1.3) (or (1.4)) exists, then
we say that $U(x)$ is stable: if for any $\epsilon>0$, there is $\delta>0$
such that when $\|u_{0}-U(x)\|_{\infty}<\delta$, we have
$\|u(t,x;u_{0})-U(x)\|_{\infty}<\epsilon$ for all $t>0$. $U(x)$ is unstable if
this not holds. $U(x)$ is asymptotically stable if for any $u_{0}(x)$, we have
$\|u(t,x;u_{0})-U(x)\|_{\infty}\to 0$ as $t\to\infty$.

###  3.1 Spectral theory of integro-differential operator

The eigenvalue problem is an important tool to study the longtime behaviors of
elliptic and parabolic problems, especially the sign of principal eigenvalue
of the linearized operator can determine the existence and stability of
nontrivial stationary solution, even the persistence of species, see [32, 44,
20, 45].

Now, we recall some known results on the principal eigenvalue of the linear
nonlocal operator $\mathfrak{L}_{\Omega,J,q}+a$, which is defined on
$C^{1}(\bar{\Omega})$ by

$(\mathfrak{L}_{\Omega,J,q}+a)[\varphi(x)]:=d\int_{\Omega}J(x-y)\varphi(y)dy-q\varphi^{\prime}(x)+a(x)\varphi(x),$

where $a(x)\in C(\bar{\Omega})\cap L^{\infty}(\Omega)$. Consider the following
eigenvalue problem:

$d\int_{\Omega}J(x-y)\varphi(y)dy-q\varphi^{\prime}(x)+a(x)\varphi(x)+\lambda\varphi(x)=0,\
\ \ x\in\Omega.$ (3.1)

The eigenvalue is called the principal eigenvalue if there is a continuous and
positive eigenfunction such that (3.1) holds. However, due to the lack of
regularity and compactness of nonlocal dispersal operator, (3.1) may not admit
principal eigenvalue. In [44], when $\Omega=\mathbb{R}$ and $a(x)$ is a
periodic function, the Krein-Rutman Theory can be used to show the existence
of principal eigenvalue of (3.1). As shown in [55, 56], if the zero order term
$a(x)$ is just assumed bounded, the generalized principal eigenvalue can be
defined to characterise the principal eigenvalue of operator
$\mathfrak{L}_{\Omega,J,q}+a$:

$\displaystyle\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a):=\sup\\{\lambda\in\mathbb{R}|\exists\varphi\in$
$\displaystyle C^{1}(\Omega)\cap C(\bar{\Omega})\ \mbox{satisfying}\
\varphi>0\ \mbox{in}\ \Omega\ \mbox{and}$ (3.2)
$\displaystyle\mathfrak{L}_{\Omega,J,q}[\varphi(x)]+a(x)\varphi(x)+\lambda\varphi(x)\leq
0\\}.$

When $q=0$, the generalized principal eigenvalue can be defined as

$\displaystyle\lambda_{p}(\mathfrak{L}_{\Omega,J}+a):=\sup\\{\lambda\in\mathbb{R}|\exists\varphi\in$
$\displaystyle C(\bar{\Omega})\ \mbox{satisfying}\ \varphi>0\ \mbox{in}\
\Omega\ \mbox{and}$
$\displaystyle\mathfrak{L}_{\Omega,J}[\varphi(x)]+a(x)\varphi(x)+\lambda\varphi(x)\leq
0\\}.$

And some standard properties have shown in [32].

######  Proposition 3.1.

Assume that $\bf{(J)}$ holds. Let $\Omega\subset\mathbb{R}$ be a non-empty
open interval, possibly unbounded. For $q\neq 0$, the following conclusions
are valid:

  1. (1)

If $\Omega_{1}\subset\Omega_{2}$ are non-empty open domains, then
$\lambda_{p}(\mathfrak{L}_{\Omega_{1},J,q}+a)\geq\lambda_{p}(\mathfrak{L}_{\Omega_{2},J,q}+a)$.

  2. (2)

If $a_{1}(x),a_{2}(x)\in C(\bar{\Omega})\cap L^{\infty}(\Omega)$ satisfying
$a_{1}(x)\leq a_{2}(x)$, then

$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a_{1})\geq\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a_{2}).$

  3. (3)

If $J$ further has a compact support and satisfies (J1), then

$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a)=\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+a).$

  4. (4)

If $\Omega$ is bounded, then $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a)$ is
continuous in $q\in(-\infty,0)\cup(0,+\infty)$.

###### Proof.

We refer to [55, Proposition 1.3], [56, Proposition 3.4] and [44, Theorem 6.2]
for the proofs of (1)-(4) when $q\neq 0$. $\Box$

######  Proposition 3.2 ([55], Theorem 1.2).

Assume that $\bf{(J)}$ holds. Let $\Omega=(l_{1},l_{2})$ be a non-empty
bounded open interval. For $q\neq 0$, we have

$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a)=\lambda_{p}^{\prime}(\mathfrak{L}_{\Omega,J,q}+a).$

If $q<0$, then

$\displaystyle\lambda_{p}^{\prime}(\mathfrak{L}_{\Omega,J,q}+a):=\inf\\{\lambda\in\mathbb{R}|\exists\varphi\in$
$\displaystyle C^{1}(\Omega)\cap C(\bar{\Omega})\ \mbox{satisfying}\
\varphi>0\ \mbox{in}\ \Omega,\varphi(l_{2})=0,\varphi(l_{1})>0,$
$\displaystyle\mbox{and}\
\mathfrak{L}_{\Omega,J,q}[\varphi(x)]+a(x)\varphi(x)+\lambda\varphi(x)\geq
0\\}.$

If $q>0$, then

$\displaystyle\lambda_{p}^{\prime}(\mathfrak{L}_{\Omega,J,q}+a):=\inf\\{\lambda\in\mathbb{R}|\exists\varphi\in$
$\displaystyle C^{1}(\Omega)\cap C(\bar{\Omega})\ \mbox{satisfying}\
\varphi>0\ \mbox{in}\ \Omega,\varphi(l_{1})=0,\varphi(l_{2})>0,$
$\displaystyle\mbox{and}\
\mathfrak{L}_{\Omega,J,q}[\varphi(x)]+a(x)\varphi(x)+\lambda\varphi(x)\geq
0\\}.$

Moreover, the infimum can be achieved. That is to say, there is a function
$\varphi\in C^{1}(\Omega)\cap C(\bar{\Omega})$ such that (3.1) holds and
$\varphi>0$ in $\Omega$ satisfying $\varphi(l_{2})=0$ for $q<0$ and
$\varphi(l_{1})=0$ for $q>0$.

######  Proposition 3.3.

Assume that $a(x)\in C(\bar{\Omega})\cap L^{\infty}(\Omega)$ is nonnegative,
$\Omega=(l_{1},l_{2})$ is a non-empty bounded open interval and
$\bf{(J),(J1)}$ holds. If further $J$ has a compact support, then for fixed
$\Omega$,

$\lambda_{p}(\mathfrak{L}_{\Omega,J,q_{1}}+a)<\lambda_{p}(\mathfrak{L}_{\Omega,J,q_{2}}+a),\
\ 0<q_{1}<q_{2};$
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q_{1}}+a)>\lambda_{p}(\mathfrak{L}_{\Omega,J,q_{2}}+a),\
\ q_{1}<q_{2}<0.$

###### Proof.

By Proposition 3.1(3), we just discuss the case for $q>0$. Note that the
eigenvalue problem (3.1) admits a principal eigen pair
$(\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a),\varphi(x))$ satisfying
$\varphi(x)>0$ in $\Omega$ and $\varphi(l_{1})=0,\varphi(l_{2})>0$.

Claim: the principal eigenfunction $\varphi(x)$ is nondecreasing. If the claim
is not true, then there exists $x_{0}\in\Omega$ such that
$\varphi^{\prime}(x_{0})<0$. Notice that

$d\int_{\Omega}J(x_{0}-y)\varphi(y)dy-q\varphi^{\prime}(x_{0})+a(x_{0})\varphi(x_{0})+\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a)\varphi(x_{0})=0,\
\ \ x\in\Omega.$

By the nonnegativity of $a(x)$ and the positivity of $\varphi(x)$, the
principal eigenvalue $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a)<0$ for any
$q>0$. However, since $\varphi\in C^{1}(\Omega)\cap C(\bar{\Omega})$ and
$\varphi(l_{1})=0,\varphi(l_{2})>0$, we can choose a nonempty set
$D\subset\Omega$ and a constant $\delta>0$ such that
$\varphi^{\prime}(x)\geq\delta$ in $D$. In view of $a(x)\in
C(\bar{\Omega})\cap L^{\infty}(\Omega)$, we can find a constant $C>0$ such
that $d\int_{\Omega}J(x-y)\varphi(y)dy+a(x)\varphi(x)\leq C$ for $x\in D$.
When $q$ is large enough satisfying $q>\frac{C}{\delta}$, there holds

$-\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a)\varphi(x)=d\int_{\Omega}J(x-y)\varphi(y)dy-q\varphi^{\prime}(x)+a(x)\varphi(x)\leq
C-q\varphi^{\prime}(x)\leq 0,x\in D$

which is a contradiction with $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+a)<0$ for
any $q>0$. The claim is true.

For $-q_{2}<0<q_{1}<q_{2}$, Proposition 3.2 means that there exist $\phi>0$
satisfying $\phi(l_{1})=0,\phi(l_{2})>0$, $\phi^{\prime}\geq 0$ and $\psi>0$
satisfying $\psi(l_{2})=0,\psi(l_{1})>0$, $\psi^{\prime}\leq 0$ such that

$\displaystyle d\int_{\Omega}J(x-y)\phi(y)dy-
q_{1}\phi^{\prime}(x)+a(x)\phi(x)+\lambda_{p}(\mathfrak{L}_{\Omega,J,q_{1}}+a)\phi(x)=0,\
\ \ x\in\Omega;$ (3.3) $\displaystyle
d\int_{\Omega}J(x-y)\psi(y)dy+q_{2}\psi^{\prime}(x)+a(x)\psi(x)+\lambda_{p}(\mathfrak{L}_{\Omega,J,-q_{2}}+a)\psi(x)=0,\
\ \ x\in\Omega.$ (3.4)

Multiplying (3.3) by $\psi$ and (3.4) by $\phi$, and then, integrating both
sides in $\Omega$ and subtracting, we obtain

$\displaystyle(\lambda_{p}(\mathfrak{L}_{\Omega,J,q_{1}}+a)-\lambda_{p}(\mathfrak{L}_{\Omega,J,-q_{2}}+a))\int_{\Omega}\phi\psi
dx$
$\displaystyle=\int_{\Omega}(q_{2}\phi\psi^{\prime}+q_{1}\phi^{\prime}\psi)dx$
$\displaystyle=q_{2}(\phi\psi)\mid_{l_{1}}^{l_{2}}+(q_{1}-q_{2})\int_{\Omega}\phi^{\prime}\psi
dx<0$

By Proposition 3.1(3),
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q_{1}}+a)<\lambda_{p}(\mathfrak{L}_{\Omega,J,-q_{2}}+a)=\lambda_{p}(\mathfrak{L}_{\Omega,J,q_{2}}+a)$.
The proof is completed.

$\Box$

######  Remark 3.4.

The following nonlocal eigenvalue problem

$d\int_{\Omega}J(x-y)[\varphi(y)-\varphi(x)]dy-q\varphi^{\prime}(x)+b(x)\varphi(x)+\lambda\varphi(x)=0,\
\ \ x\in\Omega.$

can be written as (3.1) as long as $a(x):=b(x)-d\int_{\Omega}J(x-y)dy$.

###  3.2 The existence, uniqueness and stability of nontrivial stationary
solution

Firstly, we shall the strong maximal principle for the stationary problem.

######  Lemma 3.5 (Strong maximum principle).

For any given $T>0$, assume that, for some $c(x)\in L^{\infty}(\Omega)$ and
$q>0$, function $u(x)\in C^{1}(\bar{\Omega})$ satisfies $u(x)\geq 0$ in
$\bar{\Omega}$ and

$\begin{cases}d\int_{\Omega}J(x-y)[u(y)-u(x)]dy-qu^{\prime}(x)+c(x)u(x)\leq
0,&x\in\Omega,\\\ u(l_{1})\geq 0.\end{cases}$

Then $u(x)>0$ in $(l_{1},l_{2}]$.

###### Proof.

On the contrary, we assume that there exists $x_{0}\in(l_{1},l_{2}]$ such that
$u(x_{0})=0$. If $x_{0}\in\Omega$, then we have $u^{\prime}(x_{0})=0$, and so

$d\int_{\Omega}J(x_{0}-y)u(y)dy=d[\int_{\Omega}J(x_{0}-y)u(y)-u(x_{0})]dy-
qu^{\prime}(x_{0})+c(x_{0})u(x_{0})\leq 0,$

which is impossible. If $x_{0}=l_{2}$, then $u(x)>0$ in $(l_{1},l_{2})$,
namely, $u^{\prime}(l_{2})\leq 0$. Then

$d\int_{\Omega}J(l_{2}-y)u(y)dy-
qu^{\prime}(l_{2})=d[\int_{\Omega}J(l_{2}-y)u(y)-u(l_{2})]dy-
qu^{\prime}(l_{2})+c(l_{2})u(l_{2})\leq 0,$

which is a contraction due to $q>0$. The proof is completed. $\Box$

######  Remark 3.6.

For any given $T>0$, assume that, for some $c(x)\in L^{\infty}(\Omega)$ and
$q<0$, function $u(x)\in C^{1}(\bar{\Omega})$ satisfies $u(x)\geq 0$ in
$\bar{\Omega}$ and

$\begin{cases}d\int_{\Omega}J(x-y)[u(y)-u(x)]dy-qu^{\prime}(x)+c(x)u(x)\leq
0,&x\in\Omega,\\\ u(l_{2})\geq 0.\end{cases}$

Then $u(x)>0$ in $[l_{1},l_{2})$.

In this subsection, the existence of nontrivial stationary solutions for (1.3)
or (1.4) will be proved by the upper-lower solution methods. For the
simplicity of discussion, we consider

$\begin{cases}d\int_{\Omega}J(x-y)[u(y)-u(x)]dy-
qu_{x}+uh(x,u)=0,&x\in{\Omega},\\\ u(l_{1})=0,\\\ 0\leq
u(x)\leq\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\},&x\in{\Omega}\end{cases}$
(3.5)

and

$\begin{cases}d[\int_{\Omega}J(x-y)u(y)dy-u]-qu_{x}+uh(x,u)=0,&x\in\Omega,\\\
u(l_{1})=0,\\\ 0\leq
u(x)\leq\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\},&x\in\Omega\end{cases}$ (3.6)

Notice that (3.5) (or (3.6)) has only one trivial solution $u=0$.In the
following, we still only consider the details for (3.5).

######  Theorem 3.7.

Problem (3.5) has a continuous and bounded nontrivial solution, provided that

$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)<0.$

###### Proof.

From subsection 3.1, we know that the eigenvalue problem

$d\int_{\Omega}J(x-y)\varphi(y)dy-q\varphi^{\prime}(x)+[h(x,0)-d\int_{\Omega}J(x-y)dy]\varphi(x)+\lambda\varphi(x)=0,\
\ \ x\in\Omega$ (3.7)

admits a principal eigenvalue
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)$
associated with eigenfunction $\varphi(x)\in C^{1}(\Omega)\cap
C(\bar{\Omega})$ satisfying $\varphi(x)>0$ in $(l_{1},l_{2}]$ and
$\varphi(l_{1})=0$. Let $\underline{u}(x)=\epsilon\varphi(x)$ in
$\bar{\Omega}$, where $\epsilon>0$ is a sufficiently small constant satisfying
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)+h(x,0)-h(x,\epsilon\varphi)\leq
0$. A direct calculation gives that

$\displaystyle
d\int_{\Omega}J(x-y)[\underline{u}(y)-\underline{u}(x)]dy-q\underline{u}_{x}+h(x,\underline{u})\underline{u}$
$\displaystyle=$
$\displaystyle\epsilon[d\int_{\Omega}J(x-y)[\varphi(y)-\varphi(x)]dy-q\varphi^{\prime}(x)+h(x,0)\varphi(x)]-[h(x,0)-h(x,\epsilon\varphi)]\epsilon\varphi$
$\displaystyle=$
$\displaystyle\epsilon\varphi[-\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)-h(x,0)+h(x,\epsilon\varphi)]\geq
0,$

which means that $\underline{u}$ is a lower solution of (3.5). Take a large
constant $M>\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\}$ as the upper solution
$\bar{u}=M$, then $\underline{u}\leq\bar{u}$. Denote $u(t,x;\bar{u})$ as the
solution of (1.3) with initial value $\bar{u}$. Lemma 2.6 means that

$\bar{u}\geq
u(t_{2},x;\underline{u})=u(t_{1},x;u(t_{2}-t_{1},\cdot;\underline{u}))\geq
u(t_{1},x;\underline{u})\geq\underline{u},\ \forall
0<t_{1}<t_{2},x\in\bar{\Omega},$

and

$\underline{u}\leq
u(t_{2},x;\bar{u})=u(t_{1},x;u(t_{2}-t_{1},\cdot;\bar{u}))\leq
u(t_{1},x;\bar{u})\leq\bar{u},\ \forall 0<t_{1}<t_{2},x\in\bar{\Omega}.$

Since $u(t,x;\underline{u})$ is increasing in $t$ and uniformly bounded, there
exists $U_{1}(x)\in C(\bar{\Omega})$ such that
$\lim\limits_{t\to\infty}u(t,x;\underline{u})=U_{1}(x)$ for any given
$x\in\bar{\Omega}$. Noting that $u(t,x;\underline{u})$ is continuous in
$x\in\bar{\Omega}$(Lemma 5.1), by Dini theorem, $u(t,x;\underline{u})$
converges uniformly to $U_{1}(x)$ as $t\to\infty$. Similarly, there exists
$U_{2}(x)\in C(\bar{\Omega})$ such that
$\lim\limits_{t\to\infty}u(t,x;\bar{u})=U_{2}(x)$ uniformly on $\bar{\Omega}$.
Namely, (3.5) has a continuous and bounded nontrivial solution. This completes
the proof. $\Box$

######  Theorem 3.8.

Any nontrivial solution $U(x)$ of (3.5) satisfies $U(x)\in
C^{1}(\bar{\Omega})$ and $U(x)>0$ in $(l_{1},l_{2}]$.

###### Proof.

Obviously, $0\leq U(x)\leq\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\}$. Note that
we can find a positive constant $K>\|h(x,0)\|_{\infty}$ such that

$d\int_{\Omega}J(x-y)[U(y)-U(x)]dy-
qU_{x}-KU=[-K-h(x,U)]U\leq(\|h(x,0)\|_{\infty}-K)U\leq 0.$

It follows from Lemma 3.5 that $U(x)>0$ in $(l_{1},l_{2}]$ or $U(x)\equiv 0$
in $\bar{\Omega}$. If there is $x_{0}\in(l_{1},l_{2}]$ satisfying
$U(x_{0})=0$, then it deduces $U(x)\equiv 0$ in $\bar{\Omega}$, which is a
contradiction with the definition of nontrivial solution. And so, $U(x)>0$ in
$(l_{1},l_{2}]$.

In addition, similar to the argument of [18], we prove $U_{1}(x),U_{2}(x)\in
C^{1}(\bar{\Omega})$. Without loss of generality, we only prove $U_{1}(x)\in
C^{1}(\bar{\Omega})$. Defining $w=U_{1}(x_{1})-U_{1}(x_{2})$, we know

$\displaystyle|q||U_{1}^{\prime}(x_{1})-U_{1}^{\prime}(x_{2})|\leq$
$\displaystyle
d\int_{\Omega}|J(x_{1}-y)-J(x_{2}-y)|U_{1}(y)dy+|h(x_{1},U_{1}(x_{1}))-d\int_{\Omega}J(x_{1}-y)dy||w|$
$\displaystyle+d\int_{\Omega}|J(x_{1}-y)-J(x_{2}-y)|dyU_{1}(x_{2})+|h(x_{1},U_{1}(x_{1}))-h(x_{1},U_{1}(x_{2}))|U_{1}(x_{2})$
$\displaystyle+|h(x_{1},U_{1}(x_{2}))-h(x_{2},U_{1}(x_{2}))|U_{1}(x_{2})$
$\displaystyle\leq$ $\displaystyle
Md\int_{\Omega}|J(x_{1}-y)-J(x_{2}-y)|dy+|h(x_{1},U_{1}(x_{1}))-d\int_{\Omega}J(x_{1}-y)dy||w|$
$\displaystyle+M|h(x_{1},U_{1}(x_{1}))-h(x_{1},U_{1}(x_{2}))|+M|h(x_{1},U_{1}(x_{2}))-h(x_{2},U_{1}(x_{2}))|,$

which means $U_{1}^{\prime}(x)\in C(\bar{\Omega})$ since function $h(x,u)\in
C^{1}(\bar{\Omega}\times\mathbb{R}^{+})$, $J:\mathbb{R}\to\mathbb{R}^{+}$ is
continuous and $U_{1}(x)\in C(\bar{\Omega})$. The proof is completed. $\Box$

######  Theorem 3.9.

The nontrivial solution of (3.5) is unique.

###### Proof.

To do so assume that there are two different solutions $u_{1}(x),u_{2}(x)\in
C^{1}(\bar{\Omega})$ satisfying $u_{1}(l_{1})=u_{2}(l_{1})=0$. Set
$v(x)=\sup\\{u_{1}(x),u_{2}(x)\\}$ and $b(x)=h(x,v)$. Define

$\displaystyle\lambda_{p}^{\prime\prime}(\mathfrak{L}_{\Omega,J,-q}$
$\displaystyle-d\int_{\Omega}J(x-y)dy+b):=$
$\displaystyle\inf\\{\lambda\in\mathbb{R}$ $\displaystyle|\exists\varphi\in
C^{1}(\Omega)\cap C(\bar{\Omega})\ \mbox{satisfying}\ \varphi>0\ \mbox{in}\
[l_{1},l_{2}),$ $\displaystyle\varphi(l_{2})=0,\ \mbox{and}\
\mathfrak{L}_{\Omega,J,-q}[\varphi(x)]-d\int_{\Omega}J(x-y)dy\varphi(x)+b(x)\varphi(x)+\lambda\varphi(x)\geq
0\\}.$

By [56, Proposition 6.1], we know
$\lambda_{p}^{\prime\prime}(\mathfrak{L}_{\Omega,J,-q}-d\int_{\Omega}J(x-y)dy+b)\leq
0$. If
$\lambda_{p}^{\prime\prime}(\mathfrak{L}_{\Omega,J,-q}-d\int_{\Omega}J(x-y)dy+b)=0$,
we can choose a monotone decreasing sequence
$\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ of positive numbers such that
$|\lambda_{n}|<1$ for all $n\in\mathbb{N}$ and $\lambda_{n}\to 0$ as
$n\to\infty$ and choose a nonnegative function sequence
$(\varphi_{n})_{n\in\mathbb{N}}$ such that $\varphi_{n}(l_{2})=0$,
$\|\varphi_{n}\|\leq N$ such that there holds

$\mathfrak{L}_{\Omega,J,-q}[\varphi_{n}(x)]-d\int_{\Omega}J(x-y)dy\varphi_{n}(x)+b(x)\varphi_{n}(x)+\lambda_{n}\varphi_{n}(x)\geq
0,\ \forall\ x\in\Omega.$

Then we have $0\leq\varphi_{n}(x)\leq
w:=\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\}$, and so
$(\varphi_{n})_{n\in\mathbb{N}}$ is uniformly bounded in $C([l_{1},l_{2}])$.
Obviously, $\inf_{\bar{\Omega}}\varphi^{\prime}_{n}(x)$ is achieved at some
point $x_{1}^{n}$ on $\bar{\Omega}$. Then there exists a large constant $M>0$
such that

$\displaystyle q\inf_{\bar{\Omega}}\varphi^{\prime}_{n}(x)$
$\displaystyle=q\varphi^{\prime}_{n}(x_{1}^{n})$
$\displaystyle\geq-d\int_{\Omega}J(x_{1}^{n}-y)\varphi_{n}(y)dy+d\int_{\Omega}J(x_{1}^{n}-y)dy\varphi_{n}(x_{1}^{n})-b(x_{1}^{n})\varphi_{n}(x_{1}^{n})-\lambda_{n}\varphi_{n}(x_{1}^{n})$
$\displaystyle\geq-2dw-|\lambda_{n}|w-(\sup_{u\in[0,w]}h(x,u))w$
$\displaystyle\geq-2dw-w-(\sup_{u\in[0,w]}h(x,u))w:=-M.$

By $\varphi_{n}\geq 0$ and $\varphi_{n}(l_{2})=0$, we deduce that
$(\varphi_{n})_{n\in\mathbb{N}}$ is also uniformly bounded in
$C^{1}(\bar{\Omega})$. Therefore, by diagonal extraction process, we can find
a converging subsequence such that there exists $\varphi\geq,\not\equiv 0$
such that $\varphi_{n}\to\varphi$ in $C^{1}_{loc}({\Omega})$. Moreover,
$\varphi\in L^{\infty}(\bar{\Omega})$, which satisfies $\varphi(l_{2})=0$ and

$\mathfrak{L}_{\Omega,J,-q}[\varphi(x)]-d\int_{\Omega}J(x-y)dy\varphi(x)+b(x)\varphi(x)=0,\
\forall\ x\in\Omega.$

Then, there exist sequences $(\lambda_{n})_{n\in\mathbb{N}}$ and
$(\varphi_{n})_{n\in\mathbb{N}}$ and a nonnegative nontrivial function
$\varphi\in L^{\infty}(\bar{\Omega})$ such that

  1. (1)

$\lambda_{n}>0$, $(\lambda_{n})_{n\in\mathbb{N}}$ is a monotone decreasing
sequence and converges to $0$.

  2. (2)

$(\varphi_{n})_{n\in\mathbb{N}}>0$ in $(l_{1},l_{2}]$ and $\varphi_{n}\in
C^{1}(\Omega)\cap C(\bar{\Omega})$ converges to $\varphi$ in
$C^{1}(\bar{\Omega})$ for any $x\in\bar{\Omega}$.

  3. (3)

$\varphi_{n}(l_{2})=\varphi(l_{2})=0$ and

$\displaystyle\mathfrak{L}_{\Omega,J,-q}[\varphi_{n}(x)]-d\int_{\Omega}J(x-y)dy\varphi_{n}(x)+b(x)\varphi_{n}(x)+\lambda_{n}\varphi_{n}(x)\geq
0,\ \forall\ x\in\Omega.$ (3.8)

If
$\lambda_{p}^{\prime\prime}(\mathfrak{L}_{\Omega,J,-q}-d\int_{\Omega}J(x-y)dy+b)<0$,
it follows from the definition of $\lambda_{p}^{\prime\prime}$ that there is
$\varphi>0$ in $[l_{1},l_{2})$ satisfying $\varphi\in C^{1}(\Omega)\cap
C(\bar{\Omega})$ such that $\varphi(l_{2})=0$ and

$\mathfrak{L}_{\Omega,J,-q}[\varphi(x)]-d\int_{\Omega}J(x-y)dy\varphi(x)+b(x)\varphi(x)=-\lambda_{p}^{\prime\prime}\varphi(x)\geq
0,\ \forall\ x\in\Omega.$

We can obtain $(1)-(3)$ by taking the sequence
$(\varphi_{n})_{n\in\mathbb{N}}$ defined for all $n$ by $\varphi_{n}=\varphi$,
and any decreasing sequence $(\lambda_{n})_{n\in\mathbb{N}}$ that converges to
$0$.

We multiply (3.8) by $u_{1}(x)$ and integrate it over $\Omega$, and then

$\mathcal{R}_{n}:=d\int_{\Omega}\int_{\Omega}J(x-y)(\varphi_{n}(y)-\varphi_{n}(x))dyu_{1}(x)dx+q\int_{\Omega}\varphi_{n}^{\prime}(x)u_{1}(x)dx+\int_{\Omega}(b(x)+\lambda_{n})\varphi_{n}(x)u_{1}(x)dx\geq
0.$

Thus,

$\liminf\limits_{n\to\infty}\mathcal{R}_{n}\geq 0.$

Notice that

$\displaystyle\mathcal{R}_{n}=$ $\displaystyle
d\int_{\Omega}\int_{\Omega}J(x-y)(u_{1}(y)-u_{1}(x))dy\varphi_{n}(x)dx-q\int_{\Omega}\varphi_{n}(x)u_{1}^{\prime}(x)dx+\varphi_{n}(x)u_{1}(x)|_{l_{1}}^{l_{2}}$
$\displaystyle+\int_{\Omega}(b(x)+\lambda_{n})\varphi_{n}(x)u_{1}(x)dx$
$\displaystyle=\int_{\Omega}(b(x)-h(x,u_{1}(x)))\varphi_{n}(x)u_{1}(x)dx+\int_{\Omega}\lambda_{n}\varphi_{n}(x)u_{1}(x)dx.$

Since $h(x,u)$ is decreasing in $u$ on $\mathbb{R}^{+}$ and
$v(x)\geq,\not\equiv u_{1}(x)$, we get

$\limsup\limits_{n\to\infty}\mathcal{R}_{n}=\int_{\Omega}[h(x,v(x))-h(x,u_{1}(x))]\varphi(x)u_{1}(x)dx<0.$

Therefore,

$0\leq\liminf\limits_{n\to\infty}\mathcal{R}_{n}\leq\limsup\limits_{n\to\infty}\mathcal{R}_{n}<0.$

We get a contradiction, and so $u_{1}(x)\equiv u_{2}(x)$. The proof is
completed. $\Box$

######  Theorem 3.10.

Assume that $U(x)\in C^{1}(\bar{\Omega})$ is the unique nontrivial solution of
(3.5). Then $U(x)$ is asymptotically stable.

###### Proof.

From Theorem 3.7, we can choose the upper solution
$\bar{u}=\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\}$ and lower solution
$\underline{u}=\epsilon\varphi$, where $\varphi$ is the corresponding
principal eigenfunction with principal eigenvalue
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)$ and
$\epsilon$ is small enough such that $\epsilon\varphi\leq u(1,x;u_{0})$, where
$u(1,x;u_{0})>0$. And then, for any given $u_{0}(x)\in C^{1}(\bar{\Omega})$,
the comparison argument deduces

$\underline{u}\leq u(t,x;\underline{u})\leq u(t,x;u(1,x;u_{0}))\leq
u(t,x;\bar{u})\leq\bar{u},\forall t>1.$

From the fact that $u(t,x;\underline{u})$ is increasing in $t>1$ and
$u(t,x;\bar{u})$ is decreasing in $t>1$, there exist $U_{1}(x),U_{2}(x)$ such
that $\lim\limits_{t\to\infty}u(t,x;\underline{u})=U_{1}(x)$ and
$\lim\limits_{t\to\infty}u(t,x;\bar{u})=U_{2}(x)$ uniformly on $\bar{\Omega}$.
It follows from Theorem 3.9 that $U_{1}(x)=U_{2}(x)=U(x)$ and
$\lim\limits_{t\to\infty}u(t,x;u_{0})=U(x)$ uniformly on $\bar{\Omega}$. The
proof is completed. $\Box$

######  Lemma 3.11.

Denote by $U_{0}=0$ the unique trivial solution of (3.5).

  1. (1)

If $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)<0$,
then $U_{0}$ is unstable.

  2. (2)

If $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)>0$,
then $U_{0}$ is asymptotically stable.

###### Proof.

(1) Assume that $U_{0}$ is stable. That is to say that for any $\epsilon>0$
and small enough $\delta>0$, given initial value $u_{0}(x)\in[0,\delta)$ in
$\bar{\Omega}$, we have $\|u(t,x;u_{0})\|_{\infty}<\epsilon$ for all $t>0$.
Obviously, by assumption (F1), for any $0\leq u<\epsilon$, there holds

$f(x,u)\geq(h(x,0)+\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2})u.$

Then $u(t,x;u_{0})>0$ in $(0,+\infty)\times(l_{1},l_{2}]$ by maximum
principle. And the solution of (1.3) can be written as

$u(t,x;u_{0})=u_{0}(x)+\int_{0}^{t}\\{d\int_{\Omega}J(x-y)[u(s,y)-u(s,x)]dy-
qu_{x}(s,x)+f(x,u(s,x))\\}ds.$

By Proposition 3.1(3),

$\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0)-d\int_{\Omega}J(x-y)dy)=\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy).$

Let $\varphi>0$ in $\Omega$ be the corresponding principal eigenfunction with
$\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0)-d\int_{\Omega}J(x-y)dy)$.
Proposition 3.2 means $\varphi(l_{1})>0,\varphi(l_{2})=0$. Define the inner
product $\langle\cdot,\cdot\rangle$ in $L^{2}(\Omega)$. Then

$\displaystyle\langle u(t,x;$ $\displaystyle u_{0}),\varphi\rangle$
$\displaystyle>\langle$ $\displaystyle
d\int_{0}^{t}\int_{\Omega}J(x-y)[u(s,y)-u(s,x)]dyds-\int_{0}^{t}qu_{x}(s,x)ds,\varphi\rangle+\langle\int_{0}^{t}f(x,u(s,x))ds,\varphi\rangle$
$\displaystyle\geq\langle$ $\displaystyle
d\int_{\Omega}J(x-y)[\varphi(y)-\varphi(x)]dy+q\varphi^{\prime},\int_{0}^{t}u(s,x)ds\rangle-(q\varphi\int_{0}^{t}u(s,x)ds)|_{l_{1}}^{l_{2}}$
$\displaystyle+\langle(h(x,0)+\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2})\varphi,\int_{0}^{t}u(s,x)ds\rangle$
$\displaystyle\geq\langle$
$\displaystyle-\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2}\varphi,\int_{0}^{t}u(s,x)ds\rangle-(q\varphi\int_{0}^{t}u(s,x)ds)|_{l_{1}}^{l_{2}}$
$\displaystyle\geq-$
$\displaystyle\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0))-d\int_{\Omega}J(x-y)dy}{2}\langle\varphi,\int_{0}^{t}u(s,x)ds\rangle.$

Thus, given $T>0$ large enough, for any $t>T$, there holds

$\langle
u(t,x;u_{0}),\varphi\rangle\geq-\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2}\langle\varphi,\int_{0}^{T}u(s,x)ds\rangle.$

Furthermore,

$\langle\int_{T}^{t}u(s,x;u_{0})ds,\varphi\rangle\geq-\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2}(t-T)\langle\varphi,\int_{0}^{T}u(s,x)ds\rangle,$

which implies that

$\displaystyle\langle u(t,x;u_{0}),\varphi\rangle$
$\displaystyle\geq-\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2}\langle\int_{T}^{t}u(s,x;u_{0})ds,\varphi\rangle$
$\displaystyle\geq[\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2}]^{2}(t-T)\langle\varphi,\int_{0}^{T}u(s,x)ds\rangle.$

Due to $\langle\varphi,\int_{0}^{T}u(s,x)ds\rangle>0$, we have
$\lim\limits_{t\to\infty}\langle u(t,x;u_{0}),\varphi\rangle=+\infty$. This
contradicts with $\|u(t,x;u_{0})\|_{\infty}<\epsilon$ for all $t>0$. Hence,
$U_{0}$ is unstable.
(2) Let
$(\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy),\phi)$
be the principle eigen pair of

$d\int_{\Omega}J(x-y)[\varphi(y)-\varphi(x)]dy-q\varphi^{\prime}(x)+h(x,0)\varphi(x)+\lambda\varphi(x)=0,\
\ \ x\in\Omega.$

Define
$\bar{u}=Me^{-\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2}t}\phi(x)\geq
0$ in $\bar{\Omega}$ where $M>0$ satisfying $M\phi(x)\geq u_{0}(x)$ in
$\bar{\Omega}$. Due to $h(x,u)$ decreasing in $u$, we have

$\displaystyle\bar{u}_{t}-\\{d\int_{\Omega}J(x-y)[\bar{u}(t,y)-\bar{u}(t,x)]dy-q\bar{u}_{x}+f(x,\bar{u})\\}$
$\displaystyle\geq\bar{u}_{t}-\\{d\int_{\mathbb{R}}J(x-y)[\bar{u}(t,y)-\bar{u}(t,x)]dy-q\bar{u}_{x}+\bar{u}h(x,0)\\}$
$\displaystyle\geq-\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2}\bar{u}+\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)\bar{u}$
$\displaystyle\geq\frac{\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)}{2}\bar{u}\geq
0.$

The comparison principle means $0<u(t,x;u_{0})\leq\bar{u}$. As $t\to\infty$,
we have $\lim\limits_{t\to\infty}u(t,x;u_{0})=0$, namely, $U_{0}$ is
asymptotically stable. $\Box$

######  Theorem 3.12.

If $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)\geq
0$, then (3.5) has no nontrivial solution.

###### Proof.

Case 1:
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)>0$.
Since $\bar{u}=\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\}$ is the upper solution
of (3.5), we deduce

$0<u(t,x;u_{0})\leq
u(t,x;\bar{u})\leq\bar{u}=\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\}.$

Noting that $u(t,x;\bar{u})$ is decreasing in $t$, there exists $U(x)$ such
that $\lim\limits_{t\to\infty}u(t,x;\bar{u})=U(x)$ uniformly on $\bar{\Omega}$
and $U(x)$ satisfies (3.5). If $U(x)\not\equiv 0$, then $U(x)$ is a nontrivial
solution of (3.5), and so, $U(x)$ is asymptotically stable by Theorem 3.10.
This contradicts with Lemma 3.11(2). Thus, $U(x)\equiv 0$, namely,

$\displaystyle\lim\limits_{t\to\infty}u(t,x;u_{0})=0.$ (3.9)

Assume that there exists a nontrivial solution $V(x)$ of (3.5). Theorem 3.10
implies that $\lim\limits_{t\to\infty}u(t,x;u_{0})=V(x)$, which is a
contradiction with (3.9).
Case 2:
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)=0$.
Assume that there exists a bounded nontrivial solution
$U(x)\in(0,\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\}]$ of (3.5). And then,
$U(x)$ satisfies

$d\int_{\Omega}J(x-y)[U(y)-U(x)]dy-qU_{x}+Uh(x,U)=0,x\in{\Omega}$

Denote $b(x):=h(x,U)<h(x,0)$ in $\bar{\Omega}$. By Proposition 3.1(2), we have

$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+b(x)-d\int_{\Omega}J(x-y)dy)\geq\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)=0.$

Following the proof of [56, Proposition 6.1], we deduce
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+b(x)-d\int_{\Omega}J(x-y)dy)\leq 0$.
Thus, it follows from Proposition 3.1(3) that

$\lambda_{p}(\mathfrak{L}_{\Omega,J,-q}+b(x)-d\int_{\Omega}J(x-y)dy)=\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+b(x)-d\int_{\Omega}J(x-y)dy)=0.$

By Proposition 3.2 we can check that there is $\psi\in C^{1}(\Omega)\cap
C(\overline{\Omega})$ satisfying $\psi>0$ in $\Omega$,
$\psi(l_{1})>0,\psi(l_{2})=0$, such that

$d\int_{\Omega}J(x-y)[\psi(y)-\psi(x)]dy+q\psi^{\prime}(x)+b(x)\psi(x)=0,\ \ \
x\in\Omega.$

Define the inner product $\langle\cdot,\cdot\rangle$ in $L^{2}(\Omega)$. By
calculation, we have

$\displaystyle 0=\langle$ $\displaystyle
d\int_{\Omega}J(x-y)[\psi(y)-\psi(x)]dy+q\psi^{\prime}(x)+b(x)\psi(x),U(x)\rangle$
$\displaystyle=\langle$ $\displaystyle d\int_{\Omega}J(x-y)[U(y)-U(x)]dy-
qU^{\prime}(x)+b(x)U(x),\psi(x)\rangle+(qU(x)\psi(x))|_{l_{1}}^{l_{2}}$
$\displaystyle=\langle$ $\displaystyle d[\int_{\Omega}J(x-y)[U(y)-U(x)]dy-
qU^{\prime}+h(x,U)U,\psi\rangle+(qU\psi)|_{l_{1}}^{l_{2}}$ $\displaystyle=(q$
$\displaystyle U\psi)|_{l_{1}}^{l_{2}}<0.$

Hence, we obtain a contradiction, and so the theorem holds. $\Box$

######  Lemma 3.13.

If $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)=0$,
then $U_{0}$ is asymptotically stable.

###### Proof.

Let $\varphi>0$ in $\Omega$ be the principal eigenfunction with
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)=0$.
Choose a constant $M>0$ such that $\bar{u}:=M\varphi\geq u_{0}(x)$. Similar to
Theorem 3.10, $0<u(t,x;u_{0})\leq u(t,x;\bar{u})\leq\bar{u}$ and there exists
$U(x)$ such that $\lim\limits_{t\to\infty}u(t,x;\bar{u})=U(x)$ uniformly on
$\bar{\Omega}$ and $U(x)$ satisfies (3.5). Combining with Theorem 3.12,
$U(x)\equiv 0$. Then $\lim\limits_{t\to\infty}u(t,x;u_{0})=0$. The proof is
completed. $\Box$

In summary, the results about the existence/nonexistence, uniqueness of
nontrivial solutions of (3.5) and the stability of nontrivial/trivial solution
are as follows.

######  Theorem 3.14.

For (3.5), the following results hold:

  1. (1)

If $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)<0$,
then $U_{0}$ is unstable. Moreover, (3.5) admits a unique bounded nontrivial
solution $U(x)\in C^{1}(\bar{\Omega})$ satisfying $U(x)>0$ in $(l_{1},l_{2}]$,
which is asymptotically stable.

  2. (2)

If $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)\geq
0$, then $U_{0}$ is asymptotically stable and (3.5) has no nontrivial
solution.

For (3.6), we can also have the same analysis, which looks like the proofs of
(3.5), so we still have the following theorem:

######  Theorem 3.15.

For (3.6), the following assertions are valid:

  1. (1)

If $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d<0$, then $U_{0}$ is
unstable. Moreover, (3.6) admits a unique bounded nontrivial solution $U(x)\in
C^{1}(\bar{\Omega})$ satisfying $U(x)>0$ in $(l_{1},l_{2}]$, which is
asymptotically stable.

  2. (2)

If $\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d\geq 0$, then $U_{0}$ is
asymptotically stable and (3.6) has no nontrivial solution.

##  4 The sharp criteria for persistence or extinction

######  Theorem 4.1.

Assume further that $h(x,0)\geq d\int_{\Omega}J(x-y)dy$ on $\Omega$. Then for
(1.3), there is $q_{*}>0$ such that the persistence occurs if $0<q<q_{*}$ and
the extinction occurs if $q\geq q_{*}$.

###### Proof.

Without loss of generality, we assume $q>0$. Denote the principal eigen pair
$(\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy),\varphi)$
satisfying $\varphi>0$ in $\Omega$, $\varphi(l_{1})=0,\varphi(l_{2})>0$ and

$d\int_{\Omega}J(x-y)[\varphi(y)-\varphi(x)]dy-q\varphi^{\prime}(x)+h(x,0)\varphi(x)+\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)\varphi(x)=0,\
\ \ x\in\Omega.$

Integrating both sides of the above equations in $\Omega$, we deduce

$\displaystyle-$
$\displaystyle\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)\int_{\Omega}\varphi(x)dx$
$\displaystyle=d\int_{\Omega}\int_{\Omega}J(x-y)[\varphi(y)-\varphi(x)]dydx-q\varphi|_{l_{1}}^{l_{2}}+\int_{\Omega}h(x,0)\varphi(x)dx$
$\displaystyle=\int_{\Omega}h(x,0)\varphi(x)dx-q\varphi(l_{2}).$

Since $\varphi(x)$ and $h(x,0)$ are bounded and positive,
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)<0$ for
$q\to 0$ and
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)>0$ for
$q\gg 1$. Employing Proposition 3.1(4) and Proposition 3.3, since
$h(x,0)-d\int_{\Omega}J(x-y)dy\geq 0$, we can find $q_{*}>0$ such that
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)<0$ if
$0<q<q_{*}$ and
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)\geq 0$
if $q\geq q_{*}$. Combining with Theorem 3.14, the theorem holds. $\Box$

Define

$\bar{h}=\sup_{\bar{\Omega}}h(x,0)>0,\underline{h}=\inf_{\bar{\Omega}}h(x,0);$
$q^{*}=\inf_{\mu>0}\frac{d\int_{\mathbb{R}}J(z)e^{-\mu z}dz-d+\bar{h}}{\mu}.$

By the assumption (J0) and the definition of $h(x,0)$, we can obtain from [57]
that $q^{*}\in(0,+\infty)$ and there is $\mu^{*}>0$ such that

$d\int_{\mathbb{R}}J(z)e^{-\mu^{*}z}dz-d+\bar{h}=\mu^{*}q^{*}.$

######  Theorem 4.2.

Assume further that $\inf_{\bar{\Omega}}h(x,0)>d$. Then for (1.4), the
persistence occurs if $0<q<q^{**}$ and the extinction occurs if $q\geq
q^{**}$.

###### Proof.

Firstly, set
$\bar{\lambda}=d\int_{\mathbb{R}}J(z)e^{-\mu^{*}z}dz-d+\bar{h}-q\mu^{*}$ and
$\phi_{1}(x)=e^{\mu^{*}x}$ in $\mathbb{R}$. Then $(\bar{\lambda},\phi_{1})$
satisfies

$\displaystyle
d\int_{\mathbb{R}}J(x-y)\phi_{1}(y)dy-q\phi^{\prime}_{1}+\bar{h}\phi_{1}=(\bar{\lambda}+d)\phi_{1},\
\ \mbox{on}\ \mathbb{R}.$

By the definition of (3.2), we have
$\lambda_{p}(\mathfrak{L}_{\mathbb{R},J,q}+\bar{h})\geq-(\bar{\lambda}+d)$. In
terms of Proposition 3.1(1)(2), there holds

$\lambda_{p}(\mathfrak{L}_{\mathbb{R},J,q}+\bar{h})\leq\lambda_{p}(\mathfrak{L}_{\mathbb{R},J,q}+h(x,0))\leq\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)).$

Thus,

$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d\geq-\bar{\lambda}=(q-q^{*})\mu^{*}.$

When $q\geq q^{*}$, we deduce
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d\geq 0$.

Next, set
$\underline{\lambda}=d\int_{\mathbb{R}}J(z)e^{-\mu^{*}z}dz-d+\underline{h}-q\mu^{*}$
and $\phi_{2}(x)=e^{\mu^{*}x}$ in $\Omega$. Then
$(\underline{\lambda},\phi_{2})$ satisfies

$\displaystyle
d\int_{\Omega}J(x-y)\phi_{2}(y)dy-q\phi^{\prime}_{2}+\underline{h}\phi_{2}$
$\displaystyle=(\underline{\lambda}+d)\phi_{2}-d\int_{\mathbb{R}\backslash\Omega}J(x-y)e^{\mu^{*}y}dy$
$\displaystyle\geq(\underline{\lambda}+d-d\int_{\mathbb{R}}J(z)e^{-\mu^{*}z}dz)\phi_{2},\
\ \ \ \ \ \mbox{on}\ \Omega.$

According to Proposition 3.2,
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+\underline{h})\leq-(\underline{\lambda}+d-d\int_{\mathbb{R}}J(z)e^{-\mu^{*}z}dz)$.
Since
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))\leq\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+\underline{h})$,

$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d\leq-(\underline{\lambda}-d\int_{\mathbb{R}}J(z)e^{-\mu^{*}z}dz)=q\mu^{*}+d-\underline{h}.$

When $0<q<\frac{\underline{h}-d}{\mu^{*}}<q^{*}$, we have
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d<0$. Proposition 3.1(4) and
Proposition 3.3 imply that there is $q^{**}>0$ such that
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d<0$ if $0<q<q^{**}$ and
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d\geq 0$ if $q\geq q^{**}$.

By Theorem 3.15, the solution $u(t,x;u_{0})$ of (1.4) satisfies
$\lim\limits_{t\to\infty}u(t,x;u_{0})=0$ for $q\geq q^{**}$ and
$\lim\limits_{t\to\infty}u(t,x;u_{0})=U(x)>0$ for $0<q<q^{**}$. The proof is
completed. $\Box$

######  Remark 4.3.

Noting that

$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d=\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d)\geq\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy),$

we have $q^{**}\leq q^{*}$.

##  5 The limiting behaviors of solutions with respect to advection

In this section, we analyze the limiting behaviors of positive solution to
(1.3)(or (1.4)) when $q\to 0^{+}/+\infty$. Meanwhile, if the nontrivial
stationary solution of (1.3)(or (1.4)) exists, the limiting behaviors also can
be studied. Firstly, we need to show that the positive solution of (1.3)(or
(1.4)) and the nontrivial stationary solution of (1.3)(or (1.4)) are
continuous with respect to $q$.

######  Lemma 5.1.

Let $u_{q}(t,x)$ be the unique positive solution of (1.3)(or (1.4)) for $q>0$.
Then $u_{q}(t,x)$ is continuous with respect to $q>0$ for any given $t>0$ and
$x\in\bar{\Omega}$.

###### Proof.

We only prove the conclusion for (1.3).

$\begin{cases}w_{t}(t,x)=d\int_{\Omega}J(x-y)[w(t,y)-w(t,x)]dy-
q_{1}w_{x}+(q_{2}-q_{1})(u_{q_{2}})_{x}\\\
\quad\quad\quad\quad\quad\quad\quad+u_{q_{1}}h(x,u_{q_{1}})-u_{q_{2}}h(x,u_{q_{2}}),&t>0,x\in{\Omega},\\\
w(t,l_{1})=0,&t>0,\\\ w(0,x)=0,&x\in{\Omega}.\end{cases}$

Noticing that

$\displaystyle u_{q_{1}}h(x,u_{q_{1}})-u_{q_{2}}h(x,u_{q_{2}})$
$\displaystyle=u_{q_{1}}[h(x,u_{q_{1}})-h(x,u_{q_{2}})]+(u_{q_{1}}-u_{q_{2}})h(x,u_{q_{2}})$
$\displaystyle=u_{q_{1}}\left(\int_{0}^{1}\frac{\partial
h(x,\xi)}{\partial\xi}\mid_{\xi=u_{q_{2}}+s(u_{q_{1}}-u_{q_{2}})}ds\right)w+h(x,u_{q_{2}})w$
$\displaystyle=\left(u_{q_{1}}\int_{0}^{1}\frac{\partial
h(x,\xi)}{\partial\xi}\mid_{\xi=u_{q_{2}}+s(u_{q_{1}}-u_{q_{2}})}ds+h(x,u_{q_{2}})\right)w,$

it follows from $h(x,u)\in C^{1}(\bar{\Omega}\times\mathbb{R}^{+})$ that there
exists $M>0$ such that

$|u_{q_{1}}h(x,u_{q_{1}})-u_{q_{2}}h(x,u_{q_{2}})|\leq Mw.$

Since $(u_{q_{1}})_{x}$ is uniformly bounded on $Q_{\infty}$ by (2.11), we
define $v(t,x)=(q_{1}-q_{2})e^{At}>0$, where $A$ is a positive constant
independent of $q_{1},q_{2}$ and satisfying
$A>\|(u_{q_{2}})_{x}\|_{L^{\infty}}+M$. Then

$\displaystyle v_{t}(t,x)-$
$\displaystyle\left\\{d\int_{\Omega}J(x-y)[v(t,y)-v(t,x)]dy-
q_{1}v_{x}+(q_{2}-q_{1})(u_{q_{2}})_{x}+u_{q_{1}}h(x,u_{q_{1}})-u_{q_{2}}h(x,u_{q_{2}})\right\\}$
$\displaystyle\geq Av-\|(u_{q_{2}})_{x}\|_{L^{\infty}}e^{At}(q_{1}-q_{2})-Mv$
$\displaystyle\geq[A-\|(u_{q_{2}})_{x}\|_{L^{\infty}}-M]v\geq 0.$

By comparison principle, the initial value $v(0,x)=q_{1}-q_{2}>0$ deduces

$|w(t,x)|=|u_{q_{1}}(t,x)-u_{q_{1}}(t,x)|\leq v(t,x)=(q_{1}-q_{2})e^{At},$

namely, $u_{q}(t,x)$ is continuous with respect to $q>0$ for any given $t>0$
and $x\in\bar{\Omega}$. The proof is completed. $\Box$

######  Lemma 5.2.

Assume that $I\subset(0,\infty)$ and let $u_{q}(x)$ be the unique nontrivial
stationary solution of (1.3) (or (1.4)) for $q\in I$. Then $u_{q}(x)$ is
continuous with respect to $q\in I$.

###### Proof.

The proof is inspired by [58, 34]. We only prove it for (1.3). It suffices to
show that $u_{q}(x)$ is differentiable with respect to $q\in I$. Obviously,
$u_{q}(x)>0$ on ${\Omega}$. For any $(q,\varphi)\in I\times
C^{1}(\bar{\Omega})$, define the operator

$\mathcal{L}(q,\varphi)=d\int_{\Omega}J(x-y)[\varphi(y)-\varphi(x)]dy-q\varphi^{\prime}+\varphi
h(x,\varphi).$

The corresponding linear operator with $\varphi$ at $\varphi=u_{q}$ is

$\mathcal{L}_{\varphi}(q,u_{q})v=d\int_{\Omega}J(x-y)[v(y)-v(x)]dy-
qv^{\prime}+[h(x,u_{q})+u_{q}h_{u}(x,u_{q})]v.$

Denote $\lambda$ as the principal eigenvalue of operator
$\mathcal{L}_{\varphi}(q,u_{q})$. By Proposition 3.1(3), $\lambda$ is also the
principal eigenvalue of operator $\mathcal{L}_{\varphi}(-q,u_{q})$ and can be
written as

$\displaystyle\lambda:=\inf\\{\lambda\in\mathbb{R}|\exists\varphi\in$
$\displaystyle C^{1}(\Omega)\cap C(\bar{\Omega})\ \mbox{satisfying}\
\varphi>0\ \mbox{in}\ \Omega,\varphi(l_{2})=0,\varphi(l_{1})>0,$
$\displaystyle\mbox{and}\
\mathfrak{L}_{\Omega,J,q}[\varphi(x)]+a(x)\varphi(x)+\lambda\varphi(x)\geq
0\\},$

by Proposition 3.2. That is to say, there exists a principal eigenfunction
$\phi$ satisfying $\phi(x)>0$ in $\Omega$, $\phi(l_{2})=0,\phi(l_{1})>0$ such
that

$\displaystyle
d\int_{\Omega}J(x-y)[\phi(y)-\phi(x)]dy+q\phi^{\prime}+[h(x,u_{q})+u_{q}h_{u}(x,u_{q})]\phi+\lambda\phi\geq
0\ \mbox{in}\ \Omega.$ (5.1)

In terms of $\mathcal{L}(q,u_{q})=0$, namely,

$\displaystyle d\int_{\Omega}J(x-y)[u_{q}(y)-u_{q}(x)]dy-
qu_{q}^{\prime}+u_{q}h(x,u_{q})=0,$ (5.2)

we multiply (5.1) by $u_{q}$ and (5.2) by $\phi$, and then, integrate both
sides of the above equations in $\Omega$ and subtract. Together with $h(x,u)$
decreasing in $u$, we obtain

$\displaystyle-\lambda\int_{\Omega}\phi u_{q}dx\leq q(\phi
u_{q})|_{l_{1}}^{l_{2}}+\int_{\Omega}\phi u_{q}^{2}h_{u}(x,u_{q})dx<0.$

That is to say, $\lambda>0$ and operator $\mathcal{L}_{\varphi}(q,u_{q})$ is
invertible. The implicit function theorem means that the conclusion holds.
$\Box$

######  Theorem 5.3.

Let $u_{q}(x)$ be the unique nontrivial stationary solution of (1.3) for
$q\in(0,q^{*})$. Then

$\lim\limits_{q\to 0^{+}}u_{q}(x)=\gamma(x)\ \mbox{uniformly in}\
\bar{\Omega},\ \lim\limits_{q\to(q^{*})^{-}}u_{q}(x)=0\ \mbox{uniformly in}\
\bar{\Omega},$

where $\gamma(x)$ is the unique positive solution of

$d\int_{\Omega}J(x-y)[\gamma(y)-\gamma(x)]dy+\gamma
h(x,\gamma)=0,x\in{\Omega}.$

###### Proof.

By Section 3, we know that for $q\in(0,q^{*})$, the function $u_{q}(x)$
satisfies $u_{q}(l_{1})=0$, $u_{q}(x)>0$ in $(l_{1},l_{2}]$, and

$0\leq u_{q}(x)\leq\max\\{\max_{\bar{\Omega}}u_{0}(x),N\\}:=M,$

which means $u^{\prime}_{q}(l_{1})>0$ and $u_{q}(x)$ is uniformly bounded.

We first choose a small enough $\rho>0$, and consider the case $q\geq\rho$.
Let $z(x):=(M_{1}-u_{q}^{\prime}(x))e^{-N_{1}(x-l_{1})}$ for some constant
$M_{1}\geq u^{\prime}_{q}(l_{1})$ and $N_{1}>0$. Obviously,
$z(l_{1})=M_{1}-u^{\prime}_{q}(l_{1})\geq 0$. We shall obtain $z(x)\geq 0$.
Otherwise, there is $x_{0}\in(l_{1},l_{2}]$ such that
$z(x_{0})=\inf_{[l_{1},l_{2}]}z(x)<0$. And so, $z^{\prime}(x_{0})\leq 0$.
Moreover, a simple computation implies that

$\displaystyle qz^{\prime}(x)$
$\displaystyle=-qN_{1}z(x)-(qu_{q}^{\prime}(x))^{\prime}e^{-N_{1}(x-l_{1})}$
$\displaystyle=-qN_{1}z(x)+d\int_{\Omega}J(x-y)[z(y)-z(x)]dy+k_{1}(x,u_{q})z(x)+k_{2}(x,u_{q})e^{-N_{1}(x-l_{1})}$
$\displaystyle=d\int_{\Omega}J(x-y)[z(y)-z(x)]dy+(k_{1}(x,u_{q})-qN_{1})z(x)+k_{2}(x,u_{q})e^{-N_{1}(x-l_{1})},$

where

$\displaystyle k_{1}(x,u_{q})$
$\displaystyle=du_{q}(l_{2})J(x-l_{2})+d\int_{\Omega}J_{x}(x-y)dyu_{q}-u_{q}h_{x}(x,u_{q})-M_{1}h(x,u_{q})-M_{1}u_{q}h_{u}(x,u_{q});$
$\displaystyle k_{2}(x,u_{q})$ $\displaystyle=h(x,u_{q})+u_{q}h_{u}(x,u_{q}).$

By $0<u_{q}<M$ and $h(x,u)\in C^{1}([l_{1},l_{2}]\times\mathbb{R})$, we have
that $k_{i}(x,u_{q})(i=1,2)$ is uniformly bounded. That is, there there is
$M_{2}>0$ such that $|k(x,u_{q})|\leq M_{2}$. Then

$\displaystyle 0$ $\displaystyle\geq qz^{\prime}(x_{0})$
$\displaystyle=d\int_{\Omega}J(x_{0}-y)[z(y)-z(x_{0})]dy+(k_{1}(x_{0},u_{q}(x_{0}))-qN_{1})z(x_{0})+k_{2}(x_{0},u_{q}(x_{0}))e^{-N_{1}(x_{0}-l_{1})}$
$\displaystyle\geq(M_{2}-\rho N_{1})z(x_{0})-M_{2}e^{-N_{1}(x_{0}-l_{1})}>0,$

if we choose $N_{1}>0$ large enough such that

$(M_{2}-\rho N_{1})z(x_{0})-M_{2}e^{-N_{1}(x_{0}-l_{1})}>0.$

This is a contradiction, and then $u_{q}^{\prime}(x)\leq M_{1}$ for
$q\geq\rho$. Similarly, we also have $u_{q}^{\prime}(x)\geq-M_{1}$ for
$q\geq\rho$. And so, $u_{q}^{\prime}(x)$ is uniformly bounded for $q\geq\rho$.
It follows that $u_{q}(x)$ is equi-continuous for $q\geq\rho$.

For the case $q\in[0,\rho]$, since $u_{q}(x)$ is continuous in $q\in[0,\rho]$,
it deduces that, for any $\epsilon>0$ and any given $x\in\bar{\Omega}$, there
exists $\delta_{1}(x)>0$ independent of $q$ such that for any $q\in[0,\rho]$
satisfying $|q-\rho|<\delta_{1}(x)$, we have

$|u_{q}(x)-u_{\rho}(x)|<\epsilon.$

By $x\in\bar{\Omega}$, we can choose
$\delta_{3}=\min\\{\min_{x\in\bar{\Omega}}\delta_{1}(x),\rho\\}>0$ such that
for any $q\in[0,\rho]$, we have $|q-\rho|<\delta_{3}$ and

$|u_{q}(x)-u_{\rho}(x)|<\epsilon.$

Meanwhile, since $u_{\rho}(x)$ is continuous in $x\in\bar{\Omega}$, we know
that, for any $\epsilon>0$, there exists $\delta_{2}(\rho)>0$ independent of
$q,x$ such that for any $x,y\in\bar{\Omega}$ satisfying
$|x-y|<\delta_{2}(\rho)$, we have

$|u_{\rho}(x)-u_{\rho}(y)|<\epsilon.$

Then, for any $\epsilon>0$, there exists
$\delta=\min\\{\min_{x\in\bar{\Omega}}\delta_{1}(x),\rho,\delta_{2}(\rho)\\}>0$
independent of $q,x$ such that for any $x,y\in\bar{\Omega}$ satisfying
$|x-y|<\delta$ and for any $q\in[0,\rho]$, we have

$|u_{q}(x)-u_{q}(y)|\leq|u_{q}(x)-u_{\rho}(x)|+|u_{\rho}(x)-u_{\rho}(y)|+|u_{\rho}(y)-u_{q}(y)|<3\epsilon.$

It follows that $u_{q}(x)$ is equi-continuous in $q\in[0,\rho]$.

Combining with the fact that $u_{q}(x)$ is uniformly bounded, by Ascoli-Arzela
theorem, there is a nonnegative function $\theta(x)\in C(\bar{\Omega})$ such
that

$\lim\limits_{q\to 0^{+}}u_{q}(x)=\theta(x)\ \mbox{uniformly in}\
\bar{\Omega}.$

Thus, as $q\to 0^{+}$, we have

$d\int_{\Omega}J(x-y)[\theta(y)-\theta(x)]dy+\theta h(x,\theta)=0.$

According to [32], the above equation has a unique positive solution for all
$d>0$.

On the other hand, we can also find a nonnegative function $\eta(x)\in
C(\bar{\Omega})$ such that

$\lim\limits_{q\to(q^{*})^{-}}u_{q}(x)=\eta(x)\ \mbox{uniformly in}\
\bar{\Omega}$

by Ascoli-Arzela theorem. Furthermore, $\eta(x)$ satisfies

$d\int_{\Omega}J(x-y)[\eta(y)-\eta(x)]dy-q^{*}\eta^{\prime}+\eta h(x,\eta)=0.$

However, Theorem 4.1 implies that the above equation has no nontrivial
solution when $q=q^{*}$, namely, $\eta(x)\equiv 0$ in ${\Omega}$. The proof is
completed. $\Box$

######  Theorem 5.4.

Let $u_{q}(t,x)$ be the unique positive solution of (1.3) for $q>0$. For given
$T>0$, we have

$\lim\limits_{q\to 0^{+}}u_{q}(t,x)=v^{*}(t,x)\ \mbox{uniformly in}\
[0,T]\times\bar{\Omega},\ \lim\limits_{q\to+\infty}u_{q}(t,x)=0\
\mbox{uniformly in}\ [0,T]\times\bar{\Omega},$

where $v^{*}(t,x)$ is the unique positive solution of

$\begin{cases}v^{*}_{t}(t,x)=d\int_{\Omega}J(x-y)[v^{*}(t,y)-v^{*}(t,x)]dy+v^{*}h(x,v^{*}),&0<t<T,x\in{\Omega},\\\
v^{*}(0,x)=u_{0}(x),&x\in{\Omega}.\end{cases}$ (5.3)

###### Proof.

Similar to the proof of Theorem 5.3, by the simple compact argument (Ascoli-
Arzela theorem), there exist nonnegative functions $u^{*}(t,x),w^{*}(t,x)\in
C^{1}([0,T]\times\bar{\Omega})$ such that

$\lim\limits_{q\to 0^{+}}u_{q}(t,x)=u^{*}(t,x)\ \mbox{uniformly in}\
[0,T]\times\bar{\Omega},$ $\lim\limits_{q\to+\infty}u_{q}(t,x)=w^{*}(t,x)\
\mbox{uniformly in}\ [0,T]\times\bar{\Omega}.$

Clearly,

$u_{q}(t,x)-u_{0}(x)=\int_{0}^{t}[d\int_{\Omega}J(x-y)[u_{q}(s,y)-u_{q}(s,x)]dy-q(u_{q})_{x}(s,x)+u_{q}(s,x)h(x,u_{q}(s,x))]ds.$

Since $u_{q}^{\prime}$ is bounded in $[0,T]\times\bar{\Omega}$, as $q\to
0^{+}$, it follows that

$u^{*}(t,x)-u_{0}(x)=\int_{0}^{t}[d\int_{\Omega}J(x-y)[u^{*}(s,y)-u^{*}(s,x)]dy+u^{*}(s,x)h(x,u^{*}(s,x))]ds.$

Consequently, $u^{*}(t,x)$ satisfies (5.3). Noting that (5.3) admits a unique
positive solution by [34], there holds

$\lim\limits_{q\to 0^{+}}u_{q}(t,x)=u^{*}(t,x)\ \mbox{uniformly in}\
[0,T]\times\bar{\Omega}.$

Now we consider the case for $q\to+\infty$. Taking a smooth nonnegative
function $\zeta(t)$ satisfying $\zeta(0)=\zeta(T)=0$. Then, we multiply the
first equation of (1.3) by $\zeta(t)$ and integrate over $[0,T]$, namely,

$\displaystyle\int_{0}^{T}\zeta(t)(u_{q})_{x}(t,x)dt=$
$\displaystyle\frac{d}{q}\int_{0}^{T}\int_{\Omega}J(x-y)[u_{q}(t,y)-u_{q}(t,x)]dy\zeta(t)dt+\frac{1}{q}\int_{0}^{T}u_{q}(t,x)\zeta^{\prime}(t)dt$
$\displaystyle+\frac{1}{q}\int_{0}^{T}[u_{q}(t,x)h(x,u_{q}(t,x))]\zeta(t)dt.$

Together with the boundedness of $u_{q}$ in $\overline{Q_{T}}$, as
$q\to+\infty$, we deduce

$\int_{0}^{T}\zeta(t)(w^{*})_{x}(t,x)dt=0.$

The arbitrary of $\zeta(t)$ implies that $(w^{*})_{x}(t,x)\equiv 0$ for all
$t\in[0,T]$, and it follows that

$w^{*}(t,x)=w^{*}(t)\ \mbox{for all}\ [0,T]\times\bar{\Omega}.$

Moreover, for $t\in[0,T]$, we observe

$\displaystyle\int_{0}^{t}u_{q}(s,x)ds=$
$\displaystyle\int_{0}^{t}\int_{l_{1}}^{x}(u_{q})_{x}(s,z)dzds$
$\displaystyle=$
$\displaystyle\frac{d}{q}\int_{0}^{t}\int_{l_{1}}^{x}\int_{\Omega}J(z-y)[u_{q}(s,y)-u_{q}(s,z)]dydzds$
$\displaystyle+\frac{1}{q}\int_{0}^{t}\int_{l_{1}}^{x}[u_{q}(s,z)h(z,u_{q}(s,z))]dzds-\frac{1}{q}\int_{l_{1}}^{x}[u_{q}(t,z)-u_{0}(z)]dz.$

As $q\to+\infty$, we know

$\int_{0}^{t}w^{*}(s)ds=0\ \mbox{for all}\ t\in[0,T],$

namely, $w^{*}(t)\equiv 0$ on $[0,T]$. This has proved

$\lim\limits_{q\to+\infty}u_{q}(t,x)=0\ \mbox{uniformly in}\
[0,T]\times\bar{\Omega}.$

The proof is completed. $\Box$

######  Remark 5.5.

Note that $\gamma(x)$ is the unique nontrivial stationary solution of (5.3).

Similarly, for (1.4), we have

######  Theorem 5.6.

Let $u_{q}(x)$ be the unique nontrivial stationary solution of (1.4) for
$q\in(0,q^{**})$. Then

$\lim\limits_{q\to 0^{+}}u_{q}(x)=\theta(x)\ \mbox{uniformly in}\
\bar{\Omega},\ \lim\limits_{q\to(q^{**})^{-}}u_{q}(x)=0\ \mbox{uniformly in}\
\bar{\Omega},$

where $\theta(x)$ is the unique positive solution of

$d\int_{\Omega}J(x-y)\theta(y)dy-d\theta+\theta h(x,\theta)=0,x\in\Omega.$

######  Theorem 5.7.

Let $u_{q}(t,x)$ be the unique positive solution of (1.4) for $q>0$. For given
$T>0$, we have

$\lim\limits_{q\to 0^{+}}u_{q}(t,x)=u^{*}(t,x)\ \mbox{uniformly in}\
[0,T]\times\bar{\Omega},\ \lim\limits_{q\to+\infty}u_{q}(t,x)=0\
\mbox{uniformly in}\ [0,T]\times\bar{\Omega},$

where $u^{*}(t,x)$ is the unique positive solution of

$\begin{cases}u^{*}_{t}(t,x)=d[\int_{\Omega}J(x-y)u^{*}(t,y)dy-u^{*}]+u^{*}h(x,u^{*}),&0<t<T,x\in{\Omega},\\\
u^{*}(0,x)=u_{0}(x),&x\in{\Omega}.\end{cases}$ (5.4)

######  Remark 5.8.

Note that $\theta(x)$ is the unique nontrivial stationary solution of (5.4).

##  6 Numerical simulations

In this section, we apply some numerical simulations to support theoretical
results in previous sections and observe how advection rate $q$ affects the
longtime behaviors of solutions. Take equation (1.4) with $q>0$ for example.
Then we take $\Omega=(0,5)$ and $d=0.26$. Meanwhile, define the initial
function $u_{0}(x)=sin\frac{\pi x}{5}$, kernel function
$J(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}$, and

$f(t,x,u)=(\frac{5}{2}-\frac{x^{2}}{16}-u)u.$

Example 1. A numerical sample for case $q=0.5$ is presented in Fig. 1. We
observe that species $u$ will ultimately survive and its species density
reaches a steady state as time tends to infinite in domain $\Omega=(0,5)$. And
from Fig. 1, we find that the advection rate has an impact on the steady state
of species density. A numerical sample for case $q=1$ is shown in Fig. 2. It
follows that species $u$ will extinct in domain $\Omega=(0,5)$, which implies
that its species density will tend to $0$ as time tends to infinite. Then
Figs. 1 and 2 depict the temporal dynamics of species $u$ for system (1.4).

Figure 1: Persistence of $u$ for $q=0.5$
Figure 2: Extinction of $u$ for $q=1$
Figure 3: The estimation for threshold value $q^{**}$

Notice that $h(x,0)=\frac{5}{2}-\frac{x^{2}}{16}>d$. It follows from Fig. 3
that the dependence of $q$ on stationary solution is obtained if it exists and
we get the threshold value $q^{**}\in(0.559,0.562)$ for the existence of
stationary solution. This verifies the sharp criteria for persistence or
extinction(see Theorem 4.2).

Example 2. Choosing $q=0,0.1,0.2,0.3$, by Theorem 4.2, the species will
survive and has a steady-state. Figs. 4 and 5 describe the profiles of $u$
with different $q$ at time $t=1,4$ and the profiles of $u$ with different $q$
at location $x=1,4$. We find that when the advection rate has a tendency to
$0$, the species density will close to the species density for $q=0$.
Meanwhile, Fig. 6 shows that the stationary solution will close to the
stationary solution for $q=0$ when the advection rate $q$ tends to $0$. These
phenomena explain our results for Theorems 5.6 and 5.7 with $q\to 0^{+}$.

Figure 4: The species density of $u$ with different $q$ at time $t=1,4$
Figure 5: The species density of $u$ with different $q$ at location $x=1,4$
Figure 6: The stationary solution $u(x)$ with different $q$

Example 3. By choosing $q=2,4,6,8$, we get the profiles of $u$ with different
$q$ at time $t=1,1.2$ and the profiles of $u$ with different $q$ at location
$x=1,4$; see Figs. 7 and 8. We find that when the advection rate $q$ becomes
large, the species density will have a tendency to $0$ for any time and
location, which explain the result of Theorem 5.7 with $q\to\infty$.

Figure 7: The species density of $u$ with different $q$ at time $t=1,1.2$
Figure 8: The species density of $u$ with different $q$ at location $x=1,4$

##  7 Discussion

The paper introduces a nonlocal reaction-diffusion-advection problem in a
bounded region with Dirichlet/Neumann boundary condition. Our objective is to
investigate its spatial dynamics, namely, the sharp criteria for persistence
or extinction and the limiting behaviors of solutions with respect to
advection rate. The main conclusions can be summarized as follows:

  1. (1)

The existence/nonexistence, uniqueness and stability of nontrivial stationary
solutions are obtained. To be specific:

    1. (i)

For Neumann problem (1.3) or Dirichlet problem (1.4), if it admits a bounded
nontrivial stationary solution $U(x)$, then $U(x)$ is unique and
asymptotically stable.

    2. (ii)

(1.3) exists a nontrivial stationary solution if and only if
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0)-d\int_{\Omega}J(x-y)dy)<0$.
Similarly, (1.4) exists a nontrivial stationary solution if and only if
$\lambda_{p}(\mathfrak{L}_{\Omega,J,q}+h(x,0))+d<0$.

  2. (2)

The sharp criteria with respect to advection rate for persistence-extinction
are obtained. It can be explained in biological reality. Namely, we consider
fresh water organisms that subject to directional motion in response to water
flow with a speed of $q$. There exists a critical threshold value $\tilde{q}$,
such that the species can survive only when the velocity of water flow is less
than $\tilde{q}$. If the water flow velocity exceeds $\tilde{q}$, the species
will become extinct.

  3. (3)

The limiting behaviors of solutions with respect to advection rate are
considered. It tells us that a sufficiently large directional motion will
cause the species extinction in any situations.

  4. (4)

The numerical simulations verify our theoretical proof and show that the
advection rate has a great impact on the dynamic behaviors of species.

In addition, when $f(t,x,u)=h(x,u)u$, we have a remark with sharp criteria for
persistence or extinction. For (1.3) and (1.4), notice that the prerequisite
of the sharp criteria with respect to $q$ to be true is respectively
$h(x,0)\geq d\int_{\Omega}J(x-y)dy$ and $\inf_{\bar{\Omega}}h(x,0)>d$. So,
What should we think about the case for $0\leq h(x,0)<d\int_{\Omega}J(x-y)dy$
or $0\leq\inf_{\bar{\Omega}}h(x,0)\leq d$? Furthermore, if $h(x,0)$ is sign
changed, the sharp criteria with respect to $q$ whether can be established for
(1.3) and (1.4). It is an open problem and excepts be solved in future. The
key to solving this problem is that the monotonicity of the principal
eigenvalue of operator $\mathfrak{L}_{\Omega,J,q}+a$ with respect to $q$ for
any given $a(x)\in C(\bar{\Omega})\cap L^{\infty}(\Omega)$. Here, $a(x)$ may
be negative.

## Statements and Declarations

Competing Interests We declare that the authors have no competing interests as
defined by Springer, or other interests that might be perceived to influence
the results and/or discussion reported in this paper.

## Acknowledgement

This work is supported in part by the National Natural Science Foundation of
China (No. 11871475,
12271525) and the Fundamental Research Funds for the Central Universities of
Central South University (No. CX20230218).

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