# Approximations of Random Dispersal Operators/Equations by Nonlocal Dispersal
Operators/Equations ††thanks: Partially supported by NSF grant DMS–0907752

Wenxian Shen
Department of Mathematics and Statistics
Auburn University
Auburn, AL 36849, U.S.A.

Xiaoxia Xie
Department of Applied Mathematics
Illinois Institute of Technology
Chicago, IL 60616, U.S.A

Abstract. This paper is concerned with the approximations of random dispersal
operators/equations by nonlocal dispersal operators/equations. It first proves
that the solutions of properly rescaled nonlocal dispersal initial-boundary
value problems converge to the solutions of the corresponding random dispersal
initial-boundary value problems. Next, it proves that the principal spectrum
points of nonlocal dispersal operators with properly rescaled kernels converge
to the principal eigenvalues of the corresponding random dispersal operators.
Finally, it proves that the unique positive time periodic solutions of
nonlocal dispersal KPP equations with properly rescaled kernels converge to
the unique positive time periodic solutions of the corresponding random
dispersal KPP equations.

Key words. Nonlocal dispersal, random dispersal, KPP equation, principal
eigenvalue, principal spectrum point, positive time periodic solution.

Mathematics subject classification. 35K20, 35K57, 45C05, 45J05, 92D25.

##  1 Introduction

Both random dispersal evolution equations (or reaction diffusion equations)
and nonlocal dispersal evolution equations (or differential integral
equations) are widely used to model diffusive systems in applied sciences.
Random dispersal equations of the form

$\begin{cases}\partial_{t}u(t,x)=\Delta u(t,x)+F(t,x,u),\quad&x\in D,\cr
B_{r,b}u(t,x)=0,&x\in\partial D\,\,(x\in\mathbb{R}^{N}\,\,{\rm
if}\,\,D=\mathbb{R}^{N}),\end{cases}$ (1.1)

are usually used to model diffusive systems which exhibit local internal
interactions (i.e. the movements of organisms in the systems occur randomly
between adjacent spatial locations) and have been extensively studied (see [1,
2, 3, 6, 19, 20, 24, 29, 32, 42, 46], etc.). In (1.1), the domain $D$ is
either a bounded smooth domain in $\mathbb{R}^{N}$ or $D=\mathbb{R}^{N}$. When
$D$ is a bounded domain, either $B_{r,b}u=B_{r,D}u:=u$ (in such case,
$B_{r,D}u=0$ on $\partial D$ represents homogeneous Dirichlet boundary
condition), or $B_{r,b}u=B_{r,N}u:=\frac{\partial u}{\partial{\bf n}}$ (in
such case, $B_{r,N}u=0$ on $\partial D$ represents homogeneous Neumann
boundary condition), and when $D=\mathbb{R}^{N}$, it is assumed that
$F(t,x,u)$ is periodic in $x_{j}$ with period $p_{j}$ and
$B_{r,b}u=B_{r,P}u:=u(t,x+p_{j}{\bf e_{j}})-u(t,x)$ with ${\bf
e_{j}}=(\delta_{1j},\delta_{2j},\cdots,\delta_{Nj})$ ($\delta_{ij}=0$ if
$i\not=j$ and $\delta_{ij}=1$ if $i=j$) (in such case, $B_{r,P}u=0$ in
$\mathbb{R}^{N}$ represents periodic boundary condition).

Many applied systems exhibit nonlocal internal interaction (i.e. the movements
of organisms in the systems occur between non-adjacent spatial locations).
Nonlocal dispersal evolution equations of the form

$\begin{cases}\partial_{t}u(t,x)=\nu\int_{D\cup
D_{b}}k(y-x)[u(t,y)-u(t,x)]dy+F(t,x,u),\quad&x\in\bar{D},\cr
B_{n,b}u(t,x)=0,&x\in D_{b}\,\,{\rm if}\,\,D_{b}\not=\emptyset,\end{cases}$
(1.2)

are often used to model diffusive systems which exhibit nonlocal internal
interactions and have been recently studied by many people (see [4, 7, 8, 9,
12, 13, 14, 18, 21, 26, 28, 30, 31, 44], etc.). In (1.2), $D$ is either a
smooth bounded domain of $\mathbb{R}^{N}$ or $D=\mathbb{R}^{N}$; $\nu$ is the
dispersal rate; the kernel function $k(\cdot)$ is a smooth and nonnegative
function with compact support (the size of the support reflects the dispersal
distance) and $\int_{\mathbb{R}^{N}}k(z)dz=1$. When $D$ is bounded, either
$D_{b}=D_{D}:=\mathbb{R}^{N}\backslash{\bar{D}}$ and $B_{n,b}u=B_{n,D}:=u$ (in
such case, $u=0$ on $\mathbb{R}^{N}\backslash\bar{D}$ represents homogeneous
Dirichlet type boundary condition), or $D_{b}=D_{N}:=\emptyset$ (in such case,
nonlocal diffusion takes place only in $\bar{D}$ and hence $D_{N}=\emptyset$
represents homogeneous Neumann type boundary condition); when
$D=\mathbb{R}^{N}$, it is assumed that $F(t,x+p_{j}{\bf e_{j}},u)=F(t,x,u)$,
$D_{b}=D_{P}:=\mathbb{R}^{N}$, and $B_{n,b}u=B_{n,P}u:=u(t,x+p_{j}{\bf
e_{j}})-u(t,x)$ (hence $B_{n,P}u=0$ on $\mathbb{R}^{N}$ represents periodic
boundary condition).

Observe that (1.2) with $D_{b}=D_{D}$ and $B_{n,b}u=B_{n,D}u$ can be rewritten
as

$\partial_{t}u(t,x)=\nu\left[\int_{D}k(y-x)u(t,y)dy-u(t,x)\right]+F(t,x,u),\quad
x\in\bar{D};$ (1.3)

that (1.2) with $D_{b}=D_{N}$ reduces to

$\partial_{t}u(t,x)=\nu\int_{D}k(y-x)\left[u(t,y)-u(t,x)\right]dy+F(t,x,u),\quad
x\in\bar{D};$ (1.4)

and that (1.2) with $D=D_{P}$, $F(t,x,u)$ being periodic in $x_{j}$ with
period $p_{j}$, and $B_{n,b}u=B_{n,P}u$ can be written as

$\begin{cases}\partial_{t}u(t,x)=\nu\int_{\mathbb{R}^{N}}k(y-x)\left[u(t,y)-u(t,x)\right]dy+F(t,x,u),\quad&x\in\mathbb{R}^{N},\\\
u(t,x)=u(t,x+p_{j}{\bf e_{j}}),\quad&x\in\mathbb{R}^{N}\end{cases}$ (1.5)

$(j=1,2,\cdots N)$.

A huge amount of research has been carried out toward various dynamical
aspects of random dispersal evolution equations of the form (1.1). There are
also many research works toward various dynamical aspects of nonlocal
dispersal evolution equations of the form (1.2). It has been seen that random
dispersal evolution equations with Dirichlet, or Neumann, or period boundary
condition and nonlocal dispersal evolution equations with the corresponding
boundary condition share many similar properties. For example, a comparison
principle holds for both equations. There are also many differences between
these two types of dispersal evolution equations. For example, solutions of
random dispersal evolution equations have smoothness and certain compactness
properties, but solutions of nonlocal dispersal evolution equations do not
have such properties. Nevertheless, it is expected that nonlocal dispersal
evolution equations with Dirichlet, or Neumann, or periodic boundary condition
and small dispersal distance possess similar dynamical behaviors as those of
random dispersal evolution equations with the corresponding boundary condition
and that certain dynamics of random dispersal evolution equations with
Dirichlet, or Neumann, or periodic boundary condition can be approximated by
the dynamics of nonlocal dispersal evolution equations with the corresponding
boundary condition and properly rescaled kernels. It is of great theoretical
and practical importance to investigate whether such naturally expected
properties actually hold or not.

The objective of the current paper is to investigate how the dynamics of
random dispersal operators/equations can be approximated by those of nonlocal
dispersal operators/equations from three different perspectives, that is, from
initial-boundary value problem point of view, from spectral problem point of
view, and from asymptotic behavior point of view. To this end, we assume that
$k(\cdot)$ is of the form,

$k(z)=k_{\delta}(z):=\frac{1}{\delta^{N}}k_{0}\left(\frac{z}{\delta}\right)$
(1.6)

for some $k_{0}(\cdot)$ satisfying that $k_{0}(\cdot)$ is a smooth,
nonnegative, and symmetric (in the sense that $k_{0}(z)=k_{0}(z^{\prime})$
whenever $|z|=|z^{\prime}|$) function supported on the unit ball $B(0,1)$ and
$\int_{\mathbb{R}^{N}}k_{0}(z)dz=1$, where $\delta(>0)$ is called the
dispersal distance. We also assume that

$\nu=\nu_{\delta}:=\frac{C}{\delta^{2}},$ (1.7)

where
$C=\Big{(}\frac{1}{2}\int_{\mathbb{R}^{N}}k_{0}(z)z_{N}^{2}dz\Big{)}^{-1}$.
Throughout the rest of this paper, we will distinguish the three boundary
conditions by $i=1,2,3$. Let

$X_{1}=X_{2}=\\{u(\cdot)\in C(\bar{D},\mathbb{R})\\}$

with $\|u\|_{X_{i}}=\max_{x\in\bar{D}}|u(x)|(i=1,2)$,

$X_{3}=\\{u\in C(\mathbb{R}^{N},\mathbb{R})|u(x+p_{j}{\bf e_{j}})=u(x)\\},$

with $\|u\|_{X_{3}}=\max_{x\in\mathbb{R}^{N}}|u(x)|$. Let

$X_{i}^{+}=\\{u\in X_{i}\,|\,u(x)\geq 0\\}$

($i=1,2,3$). For $u^{1}(x),u^{2}(x)\in X_{i}$, we define

$u^{1}\leq u^{2}(u^{1}\geq u^{2})\text{ if }u^{2}-u^{1}\in
X_{i}^{+}(u^{1}-u^{2}\in X_{i}^{+})$

(i=1, 2, 3). Note that $X_{1}=X_{2}$ and the introduction of $X_{2}$ is for
convenience.

First, we investigate the approximations of solutions to the initial-boundary
value problem associated to (1.1), that is,

$\begin{cases}\partial_{t}u(t,x)=\Delta u+F(t,x,u),\quad&x\in D,\cr
B_{r,b}(t,x)u=0,\quad&x\in\partial D\quad(x\in\mathbb{R}^{N}\text{ if
}D=\mathbb{R}^{N}),\cr u(s,x)=u_{0}(x),\quad&x\in\bar{D}\end{cases}$ (1.8)

by solutions to the initial-boundary value problem associated to (1.2) with
$k(\cdot)=k_{\delta}(\cdot)$ and $\nu=\nu_{\delta}$, that is,

$\begin{cases}\partial_{t}u(t,x)=\nu_{\delta}\int_{D\cup
D_{b}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+F(t,x,u),\,\ &x\in\bar{D},\cr
B_{n,b}u(t,x)=0,\quad&x\in D_{b}\,\,\,{\rm if}\,\,D_{b}\not=\emptyset,\cr
u(s,x)=u_{0}(x),\quad&x\in\bar{D},\end{cases}$ (1.9)

where $B_{r,b}=B_{r,D}$ (resp. $B_{n,b}=B_{n,D}$ and $D_{b}=D_{D}$), or
$B_{r,b}=B_{r,N}$ (resp. $D_{b}=D_{N}(=\emptyset$)), or $B_{r,b}=B_{r,P}$
(resp. $B_{n,b}=B_{n,P}$ and $D_{b}=D_{P}$). In the rest of this paper, we
assume

(H0) $D\subset\mathbb{R}^{N}$ is either a bounded $C^{2+\alpha}$ domain for
some $0<\alpha<1$ or $D=\mathbb{R}^{N}$; $k_{\delta}(\cdot)$ is as in (1.6)
and $\nu_{\delta}$ is as in (1.7); $F(t,x,u)$ is $C^{1}$ in $t\in\mathbb{R}$
and $C^{3}$ in $(x,u)\in\mathbb{R}^{N}\times\mathbb{R}$, and when
$D=\mathbb{R}^{N}$, $F$ is periodic in $x_{j}$ with period $p_{j}$, that is,
$F(t,x+p_{j}{\bf e_{j}},u)=F(t,x,u)$ for $j=1,2,\cdots,N$.

Note that, by general semigroup theory (see [22, 35]), for any
$s\in\mathbb{R}$ and any $u_{0}\in X_{i}\cap C^{1}(\bar{D})$ with
$B_{r,b}u_{0}=0$ on $\partial D$, (1.8) with $b=D$ if $i=1$, $b=N$ if $i=2$,
and $b=P$ if $i=3$ has a unique (local) solution, denoted by
$u_{i}(t,x;s,u_{0})$. Similarly, for any $s\in\mathbb{R}$ and any $u_{0}\in
X_{i}$, (1.9) with $b=D$ if $i=1$, $b=N$ if $i=2$, and $b=P$ if $i=3$ has a
unique (local) solution, denoted by $u_{i}^{\delta}(t,x;s,u_{0})$.

Among others, we prove

Theorem A. Assume that for given $1\leq i\leq 3$, $\delta_{0}>0$,
$s\in\mathbb{R}$, $T>0$, and $u_{0}\in X_{i}\cap C^{3}(\bar{D})$ with
$B_{r,b}u_{0}=0$ if $D$ is bounded ($b=D$ if $i=1$ and $b=N$ if $i=2$),
$u_{i}(t,x;s,u_{0})$ and $u_{i}^{\delta}(t,x;s,u_{0})$ exist on $[s,s+T]$ for
all $0<\delta\leq\delta_{0}$. Assume also that $\sup_{s\leq t\leq
s+T,x\in\bar{D},0<\delta\leq\delta_{0}}|u_{i}(t,x;s,u_{0})|<\infty$. Then,

$\lim_{\delta\to
0}\sup_{t\in[s,s+T]}\|u_{i}^{\delta}(t,\cdot;s,u_{0})-u_{i}(t,\cdot;s,u_{0})\|_{X_{i}}=0.$

It should be pointed out that Theorem A is the basis for the study of
approximations of various dynamics of random dispersal evolution equations by
those of nonlocal dispersal evolution equations. It should also be pointed out
that when $F(t,x,u)\equiv 0$ in (1.8) and (1.9), similar results to Theorem A
have been proved in [10] and [11] for the Dirichlet and Neumann boundary
condition cases, respectively.

Secondly, we investigate the principal eigenvalues of time periodic random
dispersal eigenvalue problems of the form

$\begin{cases}-\partial_{t}u+\Delta u+a(t,x)u=\lambda u,\quad&x\in D,\cr
B_{r,b}u=0,\quad&x\in\partial D\,\ (x\in\mathbb{R}^{N}\text{ if
}D=\mathbb{R}^{N}),\cr u(t+T,x)=u(t,x),\quad&x\in D,\end{cases}$ (1.10)

and their nonlocal counterparts of the form

$\begin{cases}-\partial_{t}u+\nu_{\delta}\int_{D\cup
D_{b}}k_{\delta}(y-x)\left[u(t,y)-u(t,x)\right]dy+a(t,x)u=\lambda
u,\quad&x\in\bar{D},\cr B_{n,b}u=0,\quad&x\in D_{b}\,\,{\rm
if}\,\,D_{b}\not=\emptyset,\cr u(t+T,x)=u(t,x),\quad&x\in\bar{D},\end{cases}$
(1.11)

where $a(t+T,x)=a(t,x)$, and when $D=\mathbb{R}^{N}$, $a(t+T,x+p_{j}{\bf
e_{j}})=a(t,x)$ for $j=1,2,\cdots,N$, and $B_{r,b}=B_{r,D}$ (resp.
$B_{n,b}=B_{n,D}$ and $D_{b}=D_{D}$), or $B_{r,b}=B_{r,N}$ (resp.
$D_{b}=D_{N}(=\emptyset)$) or $B_{r,b}=B_{r,P}$ (resp. $B_{n,b}=B_{n,P}$ and
$D_{b}=D_{P}$). We assume that $a(t,x)$ is a $C^{1}$ function in
$(t,x)\in\mathbb{R}\times\mathbb{R}^{N}$.

The eigenvalue problems of (1.10), in particular, their associated principal
eigenvalue problems, are extensively studied and quite well understood (see
[15, 16, 17, 23, 25, 27, 34, 38], etc.). For example, with any one of the
three boundary conditions, it is known that the largest real part, denoted by
$\lambda^{r}(a)$, of the spectrum set of (1.10) is an isolated algebraically
simple eigenvalue with a positive eigenfunction, and for any other $\lambda$
in the spectrum set of (1.10), $\text{Re}\lambda\leq\lambda^{r}(a)$
($\lambda^{r}(a)$ is called the principal eigenvalue of $\eqref{main-random-
eigenvalue}$ in literature).

The eigenvalue problems (1.11) have also been studied recently by many people
(see [5, 12, 27, 36, 38, 39, 40, 41], etc.). Let $\lambda^{\delta}(a)$ be the
largest real part of the spectrum set of (1.11) with any one of the three
boundary conditions. $\lambda^{\delta}(a)$ is called the principal spectrum
point of (1.11). $\lambda^{\delta}(a)$ is also called the principal eigenvalue
of (1.11), if it is an isolated algebraically simple eigenvalue with a
positive eigenfunction (see Definition 3.1 for detail). Note that
$\lambda^{\delta}(a)$ may not be an eigenvalue of (1.11) (see [12], [39] for
examples). Hence the principal eigenvalue of (1.11) may not exist. In [41],
the authors of the current paper studied the dependence of principal spectrum
points or principal eigenvalues (if exist) of nonlocal dispersal operators on
underlying parameters ($\delta,a(\cdot)$, and $\nu$) in a spatially
heterogeneous but temporally homogeneous case. However, the understanding is
still little to many interesting questions regarding the principal spectrum
points or principal eigenvalues (if exist) of (1.11). In this paper, we show
that the principal eigenvalue of (1.10) can be approximated by the principal
spectrum point of (1.11). In fact, we show

Theorem B. $\lim_{\delta\to 0}\lambda^{\delta}(a)=\lambda^{r}(a)$.

We remark that Theorem B is another basis for the study of approximations of
various dynamics of random dispersal evolution equations by those of nonlocal
dispersal evolution equations. We also remark that some necessary and
sufficient conditions are provided in [36] and [37] for $\lambda_{\delta}(a)$
to be the principal eigenvalue of (1.11). Among other, it is proved in [36,
Theorem A] and [37, Theorem 3.1] that $\lambda^{\delta}(a)$ is the principal
eigenvalue of (1.11) if and only if

$\lambda^{\delta}(a)>\max_{x\in\bar{D}}\left\\{-\frac{C}{\delta^{2}}+\frac{1}{T}\int_{0}^{T}a(t,x)dt\right\\}.$

This together with Theorem B implies the following remark.

######  Remark 1.1.

$\lambda^{\delta}(a)$ is the principal eigenvalue of (1.11), provided
$\delta\ll 1$.

Thirdly, we explore the asymptotic dynamics of the following time periodic
dispersal evolution equations,

$\begin{cases}\partial_{t}u=\Delta u+uf(t,x,u),\quad&x\in D,\cr
B_{r,b}u=0,\quad&x\in\partial D\,\ (x\in\mathbb{R}^{N}\text{ if
}D=\mathbb{R}^{N}),\end{cases}$ (1.12)

and

$\begin{cases}\partial_{t}u=\nu_{\delta}\int_{D\cup
D_{b}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+uf(t,x,u),\quad&x\in\bar{D},\cr
B_{n,b}u=0,&x\in D_{b}\,\,\,{\rm if}\,\,D_{b}\not=\emptyset,\end{cases}$
(1.13)

where $D$ is as in (H0). In the rest of this paper, we assume that

(H1) $f$ is $C^{1}$ in $t\in\mathbb{R}$ and $C^{3}$ in
$(x,u)\in\mathbb{R}^{N}\times\mathbb{R}$; $f(t,x,u)<0$ for $u\gg 1$ and
$\partial_{u}f(t,x,u)<0$ for $u\geq 0$; $f(t+T,x,u)=f(t,x,u)$; and when
$D=\mathbb{R}^{N}$, $f(t+T,x,u)=f(t,x+p_{j}{\bf e_{j}},u)=f(t,x,u)$ for
$j=1,2,\cdots,N$.

(H2) For (1.12), $\lambda^{r}(f(\cdot,\cdot,0))>0$, where
$\lambda^{r}(f(\cdot,\cdot,0))$ is the principle eigenvalue of (1.10) with
$a(t,x)=f(t,x,0)$.

(H2)δ For (1.13), $\lambda^{\delta}(f(\cdot,\cdot,0))>0$, where
$\lambda^{\delta}(f(\cdot,\cdot,0))$ is the principle spectrum point of (1.11)
with $a(t,x)=f(t,x,0)$.

Equations (1.12) and (1.13) are widely used to model population dynamics of
species exhibiting random interactions and nonlocal interactions, respectively
(see [4, 14, 33], etc. for (1.12) and [36] for (1.13)). Thanks to the
pioneering works of Fisher [20] and Kolmogorov et al. [29] on the following
special case of (1.12),

$\partial_{t}u=u_{xx}+u(1-u),\quad x\in\mathbb{R},$

(1.12) and (1.13) are referred to as Fisher type or KPP type equations.

The dynamics of (1.12) and (1.13) have been studied in many papers (see [24,
33, 45] and references therein for (1.12), and [36] and references therein for
(1.13)). With conditions (H1) and (H2), it is proved that (1.12) has exactly
two nonnegative time periodic solutions, one is $u\equiv 0$ which is unstable
and the other one, denoted by $u^{*}(t,x)$, is asymptotically stable and
strictly positive (see [45, Theorem 3.1], see also [33, Theorems 1.1, 1.3]).
Similar results for (1.13) under the assumptions (H1) and (H2)δ are proved in
[36, Theorem E]. We denote the strictly positive time periodic solution of
(1.13) by $u_{\delta}^{*}(t,x)$.

Note that, by Theorem B and Remark 1.1, (H2) implies (H2)δ when $0<\delta\ll
1$. Hence, we only assume (H2) in the following theorem. In this paper, we
show that

Theorem C. If (H1) and (H2) hold, then for any $\epsilon>0$, there exists
$\delta_{0}>0$, such that for all $0<\delta<\delta_{0}$, we have

$\sup_{t\in[0,T]}\|u_{\delta}^{*}(t,\cdot)-u^{*}(t,\cdot)\|_{C(\bar{D},\mathbb{R})}\leq\epsilon.$

Theorems A-C in the above show that many important dynamics of random
dispersal equations can be approximated by the corresponding dynamics of
nonlocal dispersal equations, which is of both great theoretical and practical
importance.

The rest of the paper is organized as follows. In section 2, we explore the
approximation of solutions of random dispersal evolution equations by the
solutions of nonlocal dispersal evolution equations and prove Theorem A. In
section 3, we investigate the approximation of principal eigenvalues of time
periodic random dispersal operators by the principal spectrum points of time
periodic nonlocal dispersal operators and prove Theorem B. We study in section
4 the approximation of the asymptotic dynamics of time periodic KPP equations
with random dispersal by the asymptotic dynamics of time periodic KPP
equations with nonlocal dispersal and prove Theorem C.

##  2 Approximation of Initial-boundary Value Problems of Random Dispersal
Equations by Nonlocal Dispersal Equations

In this section, we explore the approximation of solutions to (1.8) by the
solutions to (1.9). We first present some comparison principle for (1.8) and
(1.9). Then we prove Theorem A. Though the ideas of the proofs of Theorem A
for different types of boundary conditions are the same, different techniques
are needed for different boundary conditions. We hence give proofs of Theorem
A for different boundary conditions in different subsections.

###  2.1 Comparison principle for random and nonlocal dispersal evolution
equations

In this subsection, we present a comparison principle for random and nonlocal
evolution equations, which will be applied in the proof of Theorem A in this
section as well as in the proofs of Theorem B and C in sections 3 and 4.

######  Definition 2.1 (Super- and sub- solutions).

A continuous function $u(t,x)$ on $[s,s+T)\times\mathbb{R}^{N}$ is called a
super-solution (sub-solution) of (1.9) on $(s,s+T)$ if for any $x\in\bar{D}$,
$u(t,x)$ is differentiable on $(s,s+T)$ and satisfies that

$\begin{cases}\partial_{t}u(t,x)\geq(\leq)\nu_{\delta}\int_{D\cup
D_{b}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+F(t,x,u),\quad&x\in\bar{D},\\\
B_{n,b}u(t,x)\geq(\leq)0,&x\in D_{b}\,\,{\rm if}\,\,D_{b}\not=\emptyset,\\\
u(s,x)\geq(\leq)u_{0}(x),&x\in\bar{D},\end{cases}$

when $b=D$ or $N$, or that

$\begin{cases}\partial_{t}u(t,x)\geq(\leq)\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+F(t,x,u),\quad&x\in\mathbb{R}^{N},\\\
B_{n,b}u(t,x)=0,&x\in\mathbb{R}^{N},\\\
u(s,x)\geq(\leq)u_{0}(x),&x\in\mathbb{R}^{N},\end{cases}$

when $b=P$.

Super-solutions and sub-solutions of (1.8) on $(s,s+T)$ are defined in an
analogous way.

######  Proposition 2.1 (Comparison principle).

  * (1)

Suppose that $u^{-}(t,x)$ and $u^{+}(t,x)$ are sub-solution and super-solution
of (1.8) on $(s,s+T)$, respectively, then

$u^{-}(t,x)\leq u^{+}(t,x)\quad\forall\,\,t\in[s,s+T),\,\,x\in\bar{D}.$

  * (2)

Suppose that $u^{-}(t,x)$ and $u^{+}(t,x)$ are sub-solution and super-solution
of (1.9) on $(s,s+T)$, respectively, then

$u^{-}(t,x)\leq u^{+}(t,x)\quad\forall\,\,t\in[s,s+T),\,\,x\in\bar{D}.$

###### Proof.

(1) It follows from comparison principle for parabolic equations.

(2) It follows from [36, Proposition 3.1]. ∎

###  2.2 Proof of Theorem A in the Dirichlet boundary condition case

In this subsection, we prove Theorem A in the Dirichlet boundary case.
Throughout this subsection, we assume (H0), and $B_{r,b}u=B_{r,D}u$ in (1.8),
and $D_{b}=D_{D}(=\mathbb{R}^{N}\backslash\bar{D})$ and $B_{n,b}u=B_{n,D}u$ in
(1.9). Note that $D\cup D_{b}=\mathbb{R}^{N}$ in this case. Without loss of
generality, we assume $s=0$.

###### Proof of Theorem A in the Dirichlet boundary condition case.

Let $u_{0}\in C^{3}(\bar{D})$ with $u_{0}(x)=0$ for $x\in\partial D$. Let
$u_{1}^{\delta}(t,x)$ be the solution of (1.9) with $s=0$ and $u_{1}(t,x)$ be
the solution of (1.8) with $s=0$. Suppose that $u_{1}(t,x)$ and
$u_{1}^{\delta}(t,x)$ exist on $[0,T]$. By regularity of solutions for
parabolic equations, $u_{1}\in
C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\bar{D})\cap
C^{0,2+\alpha}([0,T]\times\bar{D})$. Let $\tilde{u}_{1}$ be an extension of
$u_{1}$ to $[0,T]\times\mathbb{R}^{N}$ satisfying that $\tilde{u}_{1}\in
C^{0,2+\alpha}([0,T]\times\mathbb{R}^{N})$. Define

$L_{\delta}(z)(t,x)=\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)[z(t,y)-z(t,x)]dy.$

Let $G(t,x)=\tilde{u}_{1}(t,x)$ for
$(t,x)\in[0,T]\times\mathbb{R}^{N}\backslash\bar{D}$. Then $\tilde{u}_{1}$
verifies

$\begin{cases}\partial_{t}\tilde{u}_{1}(t,x)=L_{\delta}(\tilde{u}_{1})(t,x)+F_{\delta}(t,x)+F(t,x,\tilde{u}_{1}(t,x)),\quad&x\in\bar{D},\,\,\,\
\quad\,t\in(0,T],\\\ \tilde{u}_{1}(t,x)=G(t,x),\quad\,\
&x\in\mathbb{R}^{N}\backslash\bar{D},t\in[0,T],\\\
\tilde{u}_{1}(0,x)=u_{0}(x),\quad&x\in\bar{D},\end{cases}$

where

$\displaystyle F_{\delta}(t,x)$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-{L}_{\delta}(\tilde{u}_{1})(t,x)$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)(\tilde{u}_{1}(t,y)-\tilde{u}_{1}(t,x))dy.$

Let $w_{1}^{\delta}=\tilde{u}_{1}-u_{1}^{\delta}$. We then have

$\begin{cases}\partial_{t}w_{1}^{\delta}(t,x)=L_{\delta}(w_{1}^{\delta})(t,x)+F_{\delta}(t,x)+a_{1}^{\delta}(t,x)w_{1}^{\delta}(t,x),\quad&x\in\bar{D},\,\,\,\quad\,\
t\in(0,T],\\\
w_{1}^{\delta}(t,x)=G(t,x),\quad&x\in\mathbb{R}^{N}\backslash\bar{D},\,t\in[0,T],\\\
w_{1}^{\delta}(0,x)=0,\quad&x\in\bar{D},\end{cases}$ (2.1)

where
$a_{1}^{\delta}(t,x)=\int_{0}^{1}F_{u}[t,x,u_{1}^{\delta}(t,x)+\theta(\tilde{u}_{1}(t,x)-u_{1}^{\delta}(t,x))]d{\theta}$.

We claim that

$\begin{cases}\sup_{t\in[0,T]}\|F_{\delta}(t,\cdot)\|_{X_{1}}=O({\delta}^{\alpha}),\cr\sup_{t\in[0,T],x\in\mathbb{R}^{N}\setminus\bar{D},{\rm
dist}(x,\partial D)\leq\delta}|G(t,x)|=O(\delta).\end{cases}$ (2.2)

In fact,

$\displaystyle\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)(\tilde{u}_{1}(t,y)-\tilde{u}_{1}(t,x))dy$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}\frac{1}{\delta^{N}}k_{0}\left(\frac{y-x}{\delta}\right)(\tilde{u}_{1}(t,y)-\tilde{u}_{1}(t,x))dy$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}k_{0}(z)(\tilde{u}_{1}(t,x+\delta
z)-\tilde{u}_{1}(t,x))dz$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}}k_{0}(z)\left[\frac{\delta^{2}z_{N}^{2}}{2!}\Delta\tilde{u}_{1}(t,x)+O(\delta^{2+\alpha})\right]dz$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\left[\nu_{\delta}\delta^{2}\int_{\mathbb{R}^{N}}k_{0}(z)\frac{z_{N}^{2}}{2}dz\right]\Delta\tilde{u}_{1}(t,x)+O(\delta^{\alpha})$
$\displaystyle=\Delta\tilde{u}_{1}(t,x)-\Delta\tilde{u}_{1}(t,x)+O(\delta^{\alpha})$
$\displaystyle=O(\delta^{\alpha})\quad\forall\,\,x\in\bar{D},$

and

$\displaystyle|G(t,x)|$ $\displaystyle=|\tilde{u}_{1}(t,x)|$
$\displaystyle\leq\sup_{t\in[0,T],x\in\mathbb{R}^{N}\setminus D,z\in\partial
D,\text{dist}(x,z)\leq\delta}|\tilde{u}_{1}(t,x)-u_{1}(t,z)|$
$\displaystyle=O(\delta)\quad\forall\,\,x\in\mathbb{R}^{N}\backslash\bar{D},\,\,{\rm
dist}(x,\partial D)\leq\delta.$

Therefore, (2.2) holds.

Next, let $\bar{w}$ be given by

$\bar{w}(t,x)=e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta,$

where
$A=\underset{t\in[0,T],x\in\bar{D},0<\delta\leq\delta_{0}}{\max}a_{1}^{\delta}(t,x)$.
By direct calculation, we have

$\begin{cases}\partial_{t}\bar{w}(t,x)=L_{\delta}(\bar{w})+a_{1}^{\delta}(t,x)\bar{w}+\bar{F}_{\delta}(t,x)\quad&x\in\bar{D},\quad\quad\,\,\
\,\,t\in(0,T],\\\
\bar{w}(t,x)=e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta,\quad&x\in\mathbb{R}^{N}\backslash\bar{D},\quad
t\in[0,T],\\\ \bar{w}(0,x)=K_{2}\delta,\quad&x\in\bar{D},\end{cases}$ (2.3)

where

$\bar{F}_{\delta}(t,x)=e^{At}K_{1}{\delta}^{\alpha}+[A-a_{1}^{\delta}(t,x)]e^{At}K_{1}{\delta}^{\alpha}t-a_{1}^{\delta}(t,x)K_{2}\delta.$

By (2.2), there are $0<\tilde{\delta}_{0}\leq\delta_{0}$ and $K_{1},K_{2}>0$
such that

$\begin{cases}F_{\delta}(t,x)\leq\bar{F}_{\delta}(t,x),\quad&x\in\bar{D},\,\,\,\,t\in[0,T],\cr
G(t,x)\leq
e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta,\quad&x\in\mathbb{R}^{N}\backslash\bar{D},\,\,{\rm
dist}(x,\partial D)\leq\delta,\,t\in[0,T],\end{cases}$ (2.4)

when $0<\delta<\tilde{\delta}_{0}$. By (2.1), (2.3), (2.4), and Proposition
2.1, we obtain

$w^{\delta}(t,x)\leq\bar{w}(t,x)=e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta\quad\forall\,x\in\bar{D},\,\,t\in[0,T]$
(2.5)

for $0<\delta<\tilde{\delta}_{0}$.

Similarly, let
$\underline{w}(t,x)=e^{At}(-K_{1}{\delta}^{\alpha}t)-K_{2}\delta$. We can
prove that for $0<\delta<\tilde{\delta}_{0}$ (by reducing $\tilde{\delta}_{0}$
if necessary),

$w^{\delta}(t,x)\geq\underline{w}(t,x)=-e^{At}(K_{1}{\delta}^{\alpha}t)-K_{2}\delta\quad\forall\,\,x\in\bar{D},\,\,t\in[0,T].$
(2.6)

By (2.5) and (2.6) we have

$|w^{\delta}(t,x)|\leq
e^{At}K_{1}{\delta}^{\alpha}t+K_{2}\delta\quad\forall\,\,x\in\bar{D},\,\,t\in[0,T],$

which implies that there is $C(T)>0$ such that for any
$0<\delta<\tilde{\delta}_{0}$,

$\sup_{t\in[0,T]}\|u_{1}(\cdot,t)-u_{1}^{\delta}(\cdot,t)\|_{X_{1}}\leq
C(T){\delta}^{\alpha}.$

Theorem A in the Dirichlet boundary condition case then follows. ∎

######  Remark 2.1.

If the homogeneous Dirichlet boundary conditions $B_{r,D}u=u=0$ on $\partial
D$ and $B_{n,D}u=u=0$ on $\mathbb{R}^{N}\backslash\bar{D}$ are changed to
nonhomogeneous Dirichlet boundary conditions $B_{r,D}u=u=g(t,x)$ on $\partial
D$ and $B_{n,D}u=u=g(t,x)$ on $\mathbb{R}^{N}\backslash\bar{D}$, Theorem A
also holds, which can be proved by the similar arguments as above.

###  2.3 Proof of Theorem A in the Neumann boundary condition case

In this subsection, we prove Theorem A in the Neumann boundary condition case.
Throughout this subsection, we assume (H0), and $B_{r,b}u=B_{r,N}u$ in (1.8),
and $D_{b}=D_{N}=\emptyset$ in (1.9). Without loss of generality, we assume
$s=0$.

We first introduce two lemmas. To this end, for given $\delta>0$ and
$d_{0}>0$, let $D_{\delta}=\\{z\in D|\mathrm{dist}(z,\partial
D)<d_{0}\delta\\}$.

######  Lemma 2.1.

Let $\theta\in
C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\times\mathbb{R}^{N})\cap
C^{0,2+\alpha}([0,T]\times\mathbb{R}^{N})$ and
$\frac{\partial\theta}{\partial{\bf n}}=h$ on $\partial D$, then for $x\in
D_{\delta}$ and $\delta$ small,

$\displaystyle\frac{1}{\delta^{2}}{\int}_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)(\theta(t,y)-\theta(t,x))dy$
$\displaystyle=\frac{1}{\delta}{\int}_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x){\bf n}(\bar{x})\cdot\frac{y-x}{\delta}h(\bar{x},t)dy$
$\displaystyle+{\int}_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)\sum_{|\beta|=2}\frac{D^{\beta}\theta}{2}(\bar{x},t)\left[\left(\frac{y-\bar{x}}{\delta}\right)^{\beta}-\left(\frac{x-\bar{x}}{\delta}\right)^{\beta}\right]dy+O(\delta^{\alpha}),$

where $\bar{x}$ is the orthogonal projection of $x$ on the boundary of $D$ so
that $\|\bar{x}-y\|\leq 2d_{0}\delta$ and ${\bf n}(\bar{x})$ is the exterior
unit normal vector of $\partial D$ at $\bar{x}$.

###### Proof.

See [10, Lemma 3]. ∎

######  Lemma 2.2.

There exist $K>0$ and $\bar{\delta}>0$ such that for $\delta<\bar{\delta}$,

$\underset{\mathbb{R}^{N}\backslash D}{\int}k_{\delta}(y-x){\bf
n}(\bar{x})\frac{y-x}{\delta}dy\geq K\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy.$

###### Proof.

See [10, Lemma 4]. ∎

###### Proof of Theorem A in the Neumann boundary condition case.

Suppose that $u_{0}\in C^{3}(\bar{D})$. Let $u_{2}^{\delta}(t,x)$ be the
solution to (1.9) with $s=0$ and $u_{2}(t,x)$ be the solution to (1.8) with
$s=0$. Assume that $u_{2}(t,x)$ and $u_{2}^{\delta}(t,x)$ exist on $[0,T]$.
Then $u_{2}\in C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\bar{D})$. Let
$\tilde{u}_{2}$ be an extension of $u_{2}$ to $[0,T]\times\mathbb{R}^{N}$
satisfying that $\tilde{u}_{2}\in
C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\mathbb{R}^{N})\cap
C^{0,2+\alpha}([0,T]\times\mathbb{R}^{N})$. Define

$L_{\delta}(z)(t,x)=\nu_{\delta}\int_{D}k_{\delta}(y-x)(z(t,y)-z(t,x))dy,$

and

$\tilde{L}_{\delta}(z)(t,x)=\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)(z(t,y)-z(t,x))dy.$

Set $w_{2}^{\delta}=u_{2}^{\delta}-\tilde{u}_{2}$. Then

$\displaystyle\partial_{t}w_{2}^{\delta}(t,x)$
$\displaystyle=\partial_{t}u_{2}^{\delta}(t,x)-\partial_{t}\tilde{u}_{2}(t,x)$
$\displaystyle=[L_{\delta}(u_{2}^{\delta})(t,x)+F(t,x,u_{2}^{\delta})]-[\Delta\tilde{u}_{2}(t,x)+F(t,x,\tilde{u}_{2})]$
$\displaystyle=L_{\delta}(w_{2}^{\delta})(t,x)+a_{2}^{\delta}(t,x)w_{2}^{\delta}(t,x)+F_{\delta}(t,x),$

where
$a_{2}^{\delta}(t,x)=\int_{0}^{1}F_{u}(t,x,\tilde{u}_{2}(t,x)+\theta(u^{\delta}_{2}(t,x)-\tilde{u}_{2}(t,x)))d\theta$
and

$F_{\delta}(t,x)=\tilde{L}_{\delta}(\tilde{u}_{2})(t,x)-\Delta\tilde{u}_{2}(t,x)-\nu_{\delta}\int_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)(\tilde{u}_{2}(t,y)-\tilde{u}_{2}(t,x))dy.$

Hence $w_{2}^{\delta}$ verifies

$\begin{cases}\partial_{t}w_{2}^{\delta}(t,x)=L_{\delta}(w_{2}^{\delta})(t,x)+a_{2}^{\delta}(t,x)w_{2}^{\delta}(t,x)+F_{\delta}(t,x),&\quad
x\in\bar{D},\\\ w_{2}^{\delta}(0,x)=0,&\quad x\in\bar{D}.\end{cases}$ (2.7)

To prove the theorem, let us pick an auxiliary function $v$ as a solution to

$\begin{cases}\partial_{t}v(t,x)=\Delta
v(t,x)+a_{2}^{\delta}(t,x)v(t,x)+h(t,x),\quad&x\in D,\,\,\,\,t\in(0,T],\\\
\frac{\partial v}{\partial{\bf n}}(t,x)=g(t,x),\quad&x\in\partial
D,\,t\in[0,T],\\\ v(0,x)=v_{0}(x),\quad&x\in D\end{cases}$

for some smooth functions $h(t,x)\geq 1$, $g(t,x)\geq 1$ and $v_{0}(x)\geq 0$
such that $v(t,x)$ has an extension $\tilde{v}(t,x)\in
C^{1+\frac{\alpha}{2},2+\alpha}((0,T]\times\mathbb{R}^{N})\cap
C^{0,2+\alpha}([0,T]\times\mathbb{R}^{N})$. Then $v$ is a solution to

$\begin{cases}\partial_{t}v(t,x)=L_{\delta}(v)(t,x)+a_{2}^{\delta}(t,x)v(t,x)+H(t,x,\delta),\quad&x\in\bar{D},t\in(0,T],\\\
v(0,x)=v_{0}(x),\quad&x\in\bar{D},t\in[0,T],\end{cases}$ (2.8)

where

$H(t,x,\delta)=\Delta\tilde{v}(t,x)-\tilde{L}_{\delta}(v)(t,x)+\nu_{\delta}\int_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)(\tilde{v}(t,y)-\tilde{v}(t,x))dy+h(t,x).$

By Lemma 2.1 and the first estimate in (2.2), we have the following estimate
for $H(x,t,\delta)$:

$\displaystyle H(t,x,\delta)$
$\displaystyle=\Delta\tilde{v}(t,x)-\tilde{L}_{\delta}(v)(t,x)+\frac{C}{\delta^{2}}\int_{\mathbb{R}^{N}\backslash
D}k_{\delta}(y-x)(\tilde{v}(t,y)-\tilde{v}(t,x))dy+h(t,x)$
$\displaystyle\geq\frac{C}{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x){\bf n}(\bar{x})\frac{y-x}{\delta}g(\bar{x},t)dy$
$\displaystyle\qquad+C\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)\sum_{|\beta|=2}\frac{D^{\beta}\tilde{v}}{2}(\bar{x},t)\left[\left(\frac{y-\bar{x}}{\delta}\right)^{\beta}-\left(\frac{x-\bar{x}}{\delta}\right)^{\beta}\right]dy+1-C_{1}\delta^{\alpha}$
$\displaystyle\geq\frac{C}{\delta}g(\bar{x},t)\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x){\bf n}(\bar{x})\frac{y-x}{\delta}dy-
D_{1}C\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy+\frac{1}{2}$ (2.9)

for some constants $D_{1}$ and $C_{1}$ and $\delta$ sufficiently small such
that $C_{1}\delta^{\alpha}\leq\frac{1}{2}$. Then Lemma 2.2 implies that there
exist $C^{\prime}>0$ and $\delta^{\prime}$ such that

$\frac{1}{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x){\bf
n}(\bar{x})\frac{y-x}{\delta}dy\geq\frac{C^{\prime}}{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy,$

if $\delta<\delta^{\prime}$. This implies that

$\displaystyle
H(x,t,\delta)\geq\left[\frac{CC^{\prime}g(\bar{x},t)}{\delta}-D_{1}\right]\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy+\frac{1}{2},$ (2.10)

if $\delta<\delta^{\prime}$.

We now estimate $F_{\delta}(t,x)$. By Lemmas 2.1, 2.2, the first estimate in
(2.2), and the fact that $\frac{\partial\tilde{u}_{2}}{\partial{\bf n}}=0$ on
$\partial D$, we have

$\displaystyle F_{\delta}(t,x)$
$\displaystyle=O(\delta^{\alpha})+\nu_{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)(\tilde{u}_{2}(t,y)-\tilde{u}_{2}(t,x))dy$
$\displaystyle=O(\delta^{\alpha})+C\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)\sum_{|\beta|=2}\frac{D^{\beta}\theta}{2}(\bar{x},t)\left[\left(\frac{y-\bar{x}}{\delta}\right)^{\beta}-\left(\frac{x-\bar{x}}{\delta}\right)^{\beta}\right]dy$
$\displaystyle\leq
C_{2}\delta^{\alpha}+D_{1}C\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy$
$\displaystyle=C_{2}\delta^{\alpha}+D_{2}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy$ (2.11)

for some $C_{2}>0$ and $D_{2}>0$. Given $\epsilon>0$, let
$v_{\epsilon}=\epsilon v$. By (2.8), $v_{\epsilon}$ satisfies

$\begin{cases}\partial_{t}v_{\epsilon}(t,x)-L_{\delta}(v_{\epsilon})(t,x)-a_{2}^{\delta}(t,x)v_{\epsilon}(t,x)=\epsilon
H(t,x,\delta),&\quad x\in\bar{D},\\\ v_{\epsilon}(0,x)=\epsilon
v_{0}(x),&\quad x\in\bar{D}.\end{cases}$ (2.12)

By (2.10) and (2.3), there exist $C_{3}>0$ and
$0<\tilde{\delta}_{0}<\delta_{0}$ such that for
$0<\delta\leq\tilde{\delta}_{0}$,

$\displaystyle F_{\delta}(t,x)\leq
C\delta^{\alpha}+D_{2}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy\leq\frac{\epsilon}{2}+\frac{C_{3}\epsilon}{\delta}\underset{\mathbb{R}^{N}\backslash
D}{\int}k_{\delta}(y-x)dy=\epsilon
H(x,t,\delta)\quad\forall\,x\in\bar{D},\,\,t\in[0,T].$ (2.13)

Then by (2.7), (2.12), (2.13), and Proposition 2.1, we have

$-M\epsilon\leq-v_{\epsilon}\leq w_{2}^{\delta}\leq v_{\epsilon}\leq
M\epsilon\quad\forall\,\,\delta\leq\tilde{\delta}_{0},$

where $\displaystyle M=\max_{t\in[0,T],x\in\bar{D}}v(t,x)$. This implies

$\sup_{t\in[0,T]}\|u_{2}^{\delta}(t,\cdot)-u_{2}(t,\cdot)\|_{X_{2}}\rightarrow
0,\quad\mathrm{as}\,\ \delta\rightarrow 0.$

Theorem A in the Neumann boundary condition is thus proved. ∎

###  2.4 Proof of Theorem A in the periodic boundary condition case

In this subsection, we prove Theorem A in the periodic boundary condition
case. Throughout this subsection, we assume (H0), $B_{r,b}u=B_{r,P}u$ in
(1.8), and $B_{n,b}u=B_{n,P}u$ in (1.9). Without loss of generality again, we
assume $s=0$.

###### Proof of Theorem A in the periodic boundary case.

Suppose that $u_{0}\in X_{3}\cap C^{3}(\mathbb{R}^{N})$. Let
$u_{3}^{\delta}(t,x)$ be the solution to (1.9) with $s=0$ and $u_{3}(t,x)$ be
the solution to (1.8) with $s=0$. Suppose that $u_{3}(t,x)$ and
$u_{3}^{\delta}(t,x)$ exist on $[0,T]$. Set
$w_{3}^{\delta}=u_{3}^{\delta}-u_{3}$. Then $w_{3}^{\delta}$ satisfies

$\begin{cases}\partial_{t}w_{3}^{\delta}(t,x)=\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)(w_{3}^{\delta}(t,y)-w_{3}^{\delta}(t,x))dy+a_{3}^{\delta}(t,x)w_{3}^{\delta}(t,x)+F_{\delta}(t,x),&x\in\mathbb{R}^{N},\,t\in(0,T],\\\
w_{3}^{\delta}(t,x)=w_{3}^{\delta}(t,x+p_{j}\mathbb{e}_{j}),&x\in\mathbb{R}^{N},\,t\in[0,T],\\\
w_{3}^{\delta}(0,x)=0,&x\in\mathbb{R}^{N},\end{cases}$ (2.14)

where
$a_{3}^{\delta}(t,x)=\int_{0}^{1}F_{u}(t,x,u_{3}(t,x)+\theta(u^{\delta}_{3}(t,x)-u_{3}(t,x)))d\theta$
and
$F_{\delta}(t,x)=\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)[u_{3}(t,y)-u_{3}(t,x)]dy-\Delta
u_{3}$. Let

$\bar{w}(t,x)=e^{At}(K_{1}{\delta}^{\alpha}t)+K_{2}\delta,$

where $\displaystyle
A=\max_{t\in[0,T],x\in\mathbb{R}^{N},0<\delta\leq\delta_{0}}a_{3}^{\delta}(t,x)$.
Applying the similar approach as in the Dirichlet boundary condition case, we
can show that there are $K_{1}>0$, $K_{2}>0$, and $\delta_{0}>0$ such that for
$0<\delta<\delta_{0}$,

$-\bar{w}(t,x)\leq
w_{3}^{\delta}(t,x)\leq\bar{w}(t,x)\quad\forall\,\,x\in\mathbb{R}^{N},\,\,t\in[0,T].$

Theorem A in the periodic boundary condition case then follows. ∎

##  3 Approximation of Principal Eigenvalues of Time Periodic Random Dispersal
Operators by Nonlocal Dispersal Operators

In this section, we investigate the approximation of principal eigenvalues of
time periodic random dispersal operators by the principal spectrum points of
time periodic nonlocal dispersal operators. We first recall some basic
properties of principal eigenvalues of time periodic random dispersal or
parabolic operators, and basic properties of principal spectrum points of time
periodic nonlocal dispersal operators. We then prove Theorem B.

###  3.1 Basic properties

In this subsection, we present basic properties of principal eigenvalues of
time periodic parabolic operators and basic properties of principal spectrum
points of time periodic nonlocal dispersal operators.

Let

$\mathcal{X}_{1}=\mathcal{X}_{2}=\\{u\in
C(\mathbb{R}\times\bar{D},\mathbb{R})|u(t+T,x)=u(t,x)\\}$

with norm
$\|u\|_{\mathcal{X}_{i}}=\sup_{t\in[0,T]}\|u(t,\cdot)\|_{X_{i}}(i=1,2)$,

$\mathcal{X}_{3}=\\{u\in
C(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})|u(t+T,x)=u(t,x+p_{j}{\bf
e_{j}})=u(t,x)\\}$

with norm $\|u\|_{\mathcal{X}_{3}}=\sup_{t\in[0,T]}\|u(t,\cdot)\|_{X_{3}}$,
and

$\mathcal{X}_{i}^{+}=\\{u\in\mathcal{X}_{i}|u(t,x)\geq 0\\}$

$(i=1,2,3)$. And for $u^{1},u^{2}\in\mathcal{X}_{i}$, we define

$u^{1}\leq u^{2}(u^{1}\geq u^{2})\text{ if
}u^{2}-u^{1}\in\mathcal{X}_{i}^{+}\,(u_{1}-u_{2}\in\mathcal{X}_{i}^{+})$

$(i=1,2,3)$. For given $a(\cdot,\cdot)\in\mathcal{X}_{i}\cap
C^{1}(\mathbb{R}\times\mathbb{R}^{N})$ , let
$L^{\delta}_{i}(a):\mathcal{D}(L^{\delta}_{i}(a))\subset\mathcal{X}_{i}\to\mathcal{X}_{i}$
be defined as follows,

$(L^{\delta}_{1}(a)u)(t,x)=-\partial_{t}u(t,x)+\nu_{\delta}\left[\int_{D}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+a(t,x)u(t,x),\quad(t,x)\in\mathbb{R}\times\bar{D},$
(3.1)
$(L^{\delta}_{2}(a)u)(t,x)=-\partial_{t}u(t,x)+\nu_{\delta}\int_{D}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+a(t,x)u(t,x),\quad(t,x)\in\mathbb{R}\times\bar{D},$
(3.2)

and

$(L^{\delta}_{3}(a)u)(t,x)=-\partial_{t}u(t,x)+\nu_{\delta}\int_{\mathbb{R}^{N}}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+a(t,x)u(t,x),\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{N}.$
(3.3)

We first recall the definition of principal spectrum points/eigenvalues of
time periodic nonlocal dispersal operators.

######  Definition 3.1.

Let

$\lambda^{\delta}_{i}(a)=\sup\\{{\rm
Re}\lambda|\lambda\in\sigma(L^{\delta}_{i}(a))\\}$

for $i=1,2,3$.

  * (1)

$\lambda^{\delta}_{i}(a)$ is called the principal spectrum point of
$L^{\delta}_{i}(a)$.

  * (2)

If $\lambda^{\delta}_{i}(a)$ is an isolated algebraically simple eigenvalue of
$L^{\delta}_{i}(a)$ with a positive eigenfunction, then
$\lambda^{\delta}_{i}(a)$ is called the principal eigenvalue of
$L^{\delta}_{i}(a)$ or it is said that $L^{\delta}_{i}(a)$ has a principal
eigenvalue.

For the time periodic random dispersal operators, let
$a(\cdot,\cdot)\in\mathcal{X}_{i}\cap C^{1}(\mathbb{R}\times\mathbb{R}^{N})$,
and $L_{i}(a):\mathcal{D}(L_{i}(a))\subset\mathcal{X}_{i}\to\mathcal{X}_{i}$
be defined as follows,

$(L_{i}(a)u)(t,x)=-\partial_{t}u(t,x)+\Delta u(t,x)+a(t,x)u(t,x)$

for $i=1,2,3$. Note that for $u\in\mathcal{D}(L_{1}(a))$, $B_{r,D}u=u=0$ on
$\partial D$ and for $u\in\mathcal{D}(L_{2}(a))$, $B_{r,N}u=\frac{\partial
u}{\partial{\bf n}}=0$ on $\partial D$. Let

$\lambda^{r}_{i}(a)=\sup\\{{\rm Re}\lambda|\lambda\in\sigma(L_{i}(a))\\}.$

It is well known that $\lambda^{r}_{i}(a)$ is an isolated algebraically simple
eigenvalue of $L_{i}(a)$ with a positive eigenfunction (see [23]) and
$\lambda^{r}_{i}(a)$ is called the principal eigenvalue of $L_{i}(a)$.

Next we derive some properties of the principal spectrum points of nonlocal
dispersal operators by using the spectral radius of the solution operators of
the associated evolution equations. To this end, for $i=1,2,3$, define
$\Phi_{i}^{\delta}(t,s;a):X_{i}\to X_{i}$ by

$(\Phi_{i}^{\delta}(t,s;a)u_{0})(\cdot)=u_{i}(t,\cdot;s,u_{0},a),\quad
u_{0}\in X_{i},$

where $u_{1}(t,\cdot;s,u_{0},a)$ is the solution to

$\partial_{t}u(t,x)=\nu_{\delta}\left[\int_{D}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+a(t,x)u(t,x),\quad
x\in\bar{D}$ (3.4)

with $u_{1}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{1}$,
$u_{2}(t,\cdot;s,u_{0},a)$ is the solution to

$\partial_{t}u(t,x)=\nu_{\delta}\int_{D}k_{\delta}(y-x)[u(t,y)-u(t,x)]dy+a(t,x)u(t,x),\quad
x\in\bar{D}$ (3.5)

with $u_{2}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{2}$, and
$u_{3}(t,\cdot;s,u_{0},a)$ is the solution to

$\partial_{t}u(t,x)=\nu_{\delta}\left[\int_{\mathbb{R}^{N}}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+a(t,x)u(t,x),\quad
x\in\mathbb{R}^{N}$ (3.6)

with $u_{3}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{3}$. By general semigroup
property, $\Phi_{i}^{\delta}(t,s;a)$ ($i=1,2,3$) is well defined.

Let $A_{1}$ be $-\Delta$ with Dirichlet boundary condition acting on
$X_{1}\cap C_{0}(D)$. Let

$X_{1}^{r}=\mathcal{D}(A_{1}^{\alpha})$ (3.7)

for some $0<\alpha<1$ such that $C^{1}(\bar{D})\subset X_{1}^{r}$ with
$\|u\|_{X_{1}^{r}}=\|A_{1}^{\alpha}u\|_{X_{1}}$. Similarly, let $A_{2}$ be
$-\Delta$ with Neumann boundary condition acting on $X_{2}$. Let

$X_{2}^{r}=X_{2}$ (3.8)

with $\|u\|_{X_{2}^{r}}=\|u\|_{X_{2}}$, and

$X_{3}^{r}=X_{3}$ (3.9)

with $\|u\|_{X_{3}^{r}}=\|u\|_{X_{3}}$. Let

$X_{i}^{r,+}=\\{u\in X_{i}^{r}|u(x)\geq 0\\}$

(i=1, 2, 3). Similarly, for $i=1,2,3$, define $\Phi_{i}(t,s;a):X_{i}^{r}\to
X_{i}^{r}$ by

$(\Phi_{i}(t,s;a)u_{0})(\cdot)=u_{i}(t,\cdot;s,u_{0},a),\quad u_{0}\in
X_{i}^{r},$

where $u_{1}(t,\cdot;s,u_{0},a)$ is the solution to

$\begin{cases}\partial_{t}u(t,x)=\Delta u(t,x)+a(t,x)u(t,x),\quad&x\in D,\\\
u(t,x)=0,&x\in\partial D\end{cases}$ (3.10)

with $u_{1}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{1}^{r}$,
$u_{2}(t,\cdot;s,u_{0},a)$ is the solution to

$\begin{cases}\partial_{t}u(t,x)=\Delta u(t,x)+a(t,x)u(t,x),\quad&x\in D,\\\
\frac{\partial u}{\partial{\bf n}}(t,x)=0,&x\in\partial D\end{cases}$ (3.11)

with $u_{2}(s,\cdot;s,u_{0},a)=u_{0}(\cdot)\in X_{2}^{r}$, and
$u_{3}(t,\cdot;s,u_{0},a)$ is the solution to

$\begin{cases}\partial_{t}u(t,x)=\Delta
u(t,x)+a(t,x)u(t,x),\quad&x\in\mathbb{R}^{N},\\\
u(t,x+p_{j}\mathbb{e_{j}})=u(t,x),&x\in\mathbb{R}^{N}\end{cases}$ (3.12)

with $u_{3}(s,\cdot;s,u_{3},a)=u_{0}(\cdot)\in X_{3}^{r}$.

Let $r(\Phi_{i}^{\delta}(T,0;a))$ be the spectral radius of
$\Phi_{i}^{\delta}(T,0;a)$ and $\lambda_{i}^{\delta}(a)$ be the principal
spectrum point of $L_{i}^{\delta}(a)$. We have the following propositions.

######  Proposition 3.1.

Let $1\leq i\leq 3$ be given. Then

$r(\Phi_{i}^{\delta}(T,0;a))=e^{\lambda_{i}^{\delta}(a)T}.$

###### Proof.

See [41, Proposition 3.3]. ∎

Similarly, let $r(\Phi_{i}(T,0;a))$ be the spectral radius of
$\Phi_{i}(T,0;a)$ and $\lambda_{i}^{r}(a)$ be the principal eigenvalue of
$L_{i}(a)$. Note that $X_{i}^{r}$ is a strongly ordered Banach space with the
positive cone $C=\\{u\in X_{i}^{r}\,|\,u(x)\geq 0\\}$ and by the regularity, a
priori estimates, and comparison principle for parabolic equations,
$\Phi_{i}(T,0;a):X_{i}^{r}\to X_{i}^{r}$ is strongly positive and compact.
Then by the Kreĭn-Rutman Theorem (see [43]), $r(\Phi_{i}(T,0;a))$ is an
isolated algebraically simple eigenvalue of $\Phi_{i}(T,0;a)$ with a positive
eigenfunction $u_{i}^{r}(\cdot)$ and for any
$\mu\in\sigma(\Phi_{i}(T,0;a))\setminus\\{r(\Phi_{i}(T,0;a))\\}$,

$\text{Re}\mu<r(\Phi_{i}(T,0;a)).$

The following propositions then follow.

######  Proposition 3.2.

Let $1\leq i\leq 3$ be given. Then

$r(\Phi_{i}(T,0;a))=e^{\lambda_{i}^{r}(a)T}.$

Moreover, there is a codimension one subspace $Z_{i}$ of $X_{i}^{r}$ such that

$X_{i}^{r}=Y_{i}\oplus Z_{i},$

where $Y_{i}={\rm span}\\{u_{i}^{r}(\cdot)\\}$, and there are $M>0$ and
$\gamma>0$ such that for any $w_{i}\in Z_{i}$, there holds

$\frac{\|\Phi_{i}(nT,0;a)w_{i}\|_{X_{i}^{r}}}{\|\Phi_{i}(nT,0;a)u_{i}^{r}\|_{X_{i}^{r}}}\leq
Me^{-\gamma nT}.$

######  Proposition 3.3.

For given $1\leq i\leq 3$ and $a_{1},a_{2}\in\mathcal{X}_{i}\cap
C^{1}(\mathbb{R}\times\mathbb{R}^{N})$,

$|\lambda_{i}^{\delta}(a_{1})-\lambda_{i}^{\delta}(a_{2})|\leq\max_{t\in[0,T],x\in\bar{D}}|a_{1}(t,x)-a_{2}(t,x)|,$
(3.13)

and

$|\lambda_{i}^{r}(a_{1})-\lambda_{i}^{r}(a_{2})|\leq\max_{t\in[0,T],x\in\bar{D}}|a_{1}(t,x)-a_{2}(t,x)|.$
(3.14)

###### Proof.

Let $a_{0}=\max_{t\in[0,T],x\in\bar{D}}|a_{1}(t,x)-a_{2}(t,x)|$ and

$a_{1}^{\pm}(t,x)=a_{1}(t,x)\pm a_{0}.$

It is not difficult to see that

$\Phi_{i}^{\delta}(t,s;a_{1}^{\pm})=e^{\pm
a_{0}(t-s)}\Phi_{i}^{\delta}(t,s;a_{1}).$

It then follows that

$r(\Phi_{i}^{\delta}(T,0;a_{1}^{\pm}))=e^{(\lambda_{i}^{\delta}(a_{1})\pm
a_{0})T}.$ (3.15)

Observe that by Proposition 2.1, for any $u_{0}\in X_{i}^{+}$,

$\Phi_{i}^{\delta}(T,0;a_{1}^{-})u_{0}\leq\Phi_{i}^{\delta}(T,0;a_{2})u_{0}\leq\Phi_{i}^{\delta}(T,0;a_{1}^{+})u_{0}.$

This implies that

$r(\Phi_{i}^{\delta}(T,0;a_{1}^{-}))\leq r(\Phi_{i}^{\delta}(T,0;a_{2}))\leq
r(\Phi_{i}^{\delta}(T,0;a_{1}^{+})).$

This together with (3.15) implies that

$\lambda_{i}^{\delta}(a_{1})-a_{0}\leq\lambda_{i}^{\delta}(a_{2})\leq\lambda_{i}^{\delta}(a_{1})+a_{0},$
(3.16)

that is, (3.13) holds.

Similarly, we can prove that (3.14) holds. ∎

###  3.2 Proof of Theorem B in the Dirichlet boundary condition case

In this subsection, we prove Theorem B in the Dirichlet boundary condition
case. Throughout this subsection, we assume $B_{r,b}u=B_{r,D}u$ in (1.10), and
$D_{b}=D_{D}(=\mathbb{R}^{N}\setminus\bar{D})$ and $B_{n,b}u=B_{n,D}u$ in
(1.11). Note that for any $a\in\mathcal{X}_{1}\cap
C^{1}(\mathbb{R}\times\mathbb{R}^{N})$, there are $a_{n}\in\mathcal{X}_{1}\cap
C^{3}(\mathbb{R}\times\mathbb{R}^{N})$ such that
$\sup_{t\in[0,T]}\|a_{n}(t,\cdot)-a(t,\cdot)\|_{X_{1}}\to 0$ as $n\to\infty$.
By Proposition 3.3, without loss of generality, we may assume that
$a\in\mathcal{X}_{1}\cap C^{3}(\mathbb{R}\times\mathbb{R}^{N})$.

###### Proof of Theorem B in the Dirichlet boundary condition case.

First of all, for the simplicity in notation, we put

$\Phi(T,0)=\Phi_{1}(T,0;a),\,\ \quad\lambda_{1}^{r}=\lambda_{1}^{r}(a),$

and

$\Phi^{\delta}(T,0)=\Phi_{1}^{\delta}(T,0;a),\quad\lambda_{1}^{\delta}=\lambda_{1}^{\delta}(a).$

Let $u^{r}(\cdot)$ be a positive eigenfunction of $\Phi(T,0)$ corresponding to
$r(\Phi(T,0))$. Without loss of generality, we assume that
$\|u^{r}\|_{X_{1}^{r}}=1$.

We first show that for any $\epsilon>0$, there is $\delta_{1}>0$ such that for
$0<\delta<\delta_{1}$,

$\lambda_{1}^{\delta}\geq\lambda_{1}^{r}-\epsilon.$ (3.17)

In order to do so, choose $D_{0}\subset\subset D$ and $u_{0}\in X_{1}^{r}\cap
C^{3}(\bar{D})$ such that $u_{0}(x)=0$ for $x\in D\backslash D_{0}$, and
$u_{0}(x)>0$ for $x\in\text{Int}D_{0}$. By Proposition 3.2, there exist
$\alpha>0$, $M>0$, and $u^{\prime}\in Z_{1}$, such that

$u_{0}(x)=\alpha u^{r}(x)+u^{\prime}(x),$ (3.18)

and

$\|\Phi(nT,0)u^{\prime}\|_{X_{1}^{r}}\leq Me^{-\gamma
nT}e^{\lambda_{1}^{r}nT}.$ (3.19)

By Theorem A, there is $\delta_{0}>0$ such that for $0<\delta<\delta_{0}$,
there hold

$\big{(}\Phi^{\delta}(nT,0)u^{r}\big{)}(x)\geq\big{(}\Phi(nT,0)u^{r}\big{)}(x)-C^{1}(nT,\delta),$
(3.20)

and

$\big{(}\Phi^{\delta}(nT,0)u^{\prime}\big{)}(x)\leq\big{(}\Phi(nT,0)u^{\prime}\big{)}(x)+C^{2}(nT,\delta),$
(3.21)

where $C^{i}(nT,\delta)\to 0$ as $\delta\to 0$ ($i=1,2$). Hence for
$0<\delta<\delta_{0}$,

$\displaystyle\big{(}\Phi^{\delta}(nT,0)u_{0}\big{)}(x)=$
$\displaystyle\alpha\big{(}\Phi^{\delta}(nT,0)u^{r}\big{)}(x)+\big{(}\Phi^{\delta}(nT,0)u^{\prime}\big{)}(x)$
$\displaystyle\geq$
$\displaystyle\alpha\big{(}\Phi(nT,0)u^{r}\big{)}(x)-\alpha
C^{1}(nT,\delta)-C^{2}(nT,\delta)-\|\Phi(nT,0)u^{\prime}\|_{X_{1}^{r}}$
$\displaystyle\geq$ $\displaystyle\alpha e^{\lambda_{1}^{r}nT}u^{r}(x)-\alpha
C^{1}(nT,\delta)-C^{2}(nT,\delta)-Me^{-\gamma nT}e^{\lambda_{1}^{r}nT}$
$\displaystyle=$ $\displaystyle e^{(\lambda_{1}^{r}-\epsilon)nT}e^{\epsilon
nT}(\alpha u^{r}(x)-Me^{-\gamma nT})-\alpha
C^{1}(nT,\delta)-C^{2}(nT,\delta).$ (3.22)

Note that there exists $m>0$ such that

$u^{r}(x)\geq m>0\quad\text{ for }x\in\bar{D}_{0}.$

Hence for any $0<\epsilon<\gamma$, there is $n_{1}>0$ such that for $n\geq
n_{1}$,

$e^{\epsilon nT}(\alpha u^{r}(x)-Me^{-\gamma nT})\geq u_{0}(x)+1\quad{\rm
for}\quad x\in\bar{D}_{0},$ (3.23)

and there is $\delta_{1}\leq\delta_{0}$ such that for $0<\delta<\delta_{1}$,

$C^{1}(n_{1}T,\delta)+C^{2}(n_{1}T,\delta)\leq
e^{(\lambda_{1}^{r}-\epsilon)n_{1}T}.$ (3.24)

Note that $u_{0}(x)=0$ for $x\in D\backslash D_{0}$ and
$\big{(}\Phi^{\delta}(n_{1}T,0)u_{0}\big{)}(x)\geq 0$ for all $x\in\bar{D}$.
This together with (3.2)-(3.24) implies that for $\delta<\delta_{1}$,

$\big{(}\Phi^{\delta}(n_{1}T,0)u_{0}\big{)}(x)\geq
e^{(\lambda_{1}^{r}-\epsilon)n_{1}T}u_{0}(x),\quad x\in\bar{D}.$ (3.25)

By (3.25) and Proposition 2.1, for any $0<\delta<\delta_{1}$ and $n\geq 1$,

$(\Phi^{\delta}(nn_{1}T,0)u_{0})(\cdot)\geq
e^{(\lambda_{1}^{r}-\epsilon)nn_{1}T}u_{0}(\cdot).$

This together with Proposition 3.1 implies that for $0<\delta<\delta_{1}$,

$e^{\lambda_{1}^{\delta}T}=r(\Phi^{\delta}(T,0))\geq
e^{(\lambda_{1}^{r}-\epsilon)T}.$

Hence (3.17) holds.

Next, we prove that for any $\epsilon>0$, there is $\delta_{2}>0$ such that
for $0<\delta<\delta_{2}$,

$\lambda_{1}^{\delta}\leq\lambda_{1}^{r}+\epsilon.$ (3.26)

To this end, first, choose a sequence of smooth domains $\\{D_{m}\\}$ such
that $D_{1}\supset D_{2}\supset D_{3}\cdots\supset
D_{m}\supset\cdots\supset\bar{D}$, and $\cap_{m=1}^{\infty}D_{m}=\bar{D}$.
Consider the following evolution equation

$\begin{cases}\partial_{t}u(t,x)=\Delta u(t,x)+a(t,x)u(t,x),\quad&x\in
D_{m},\cr u(t,x)=0,\quad&x\in\partial D_{m}.\end{cases}$ (3.27)

Let

$X_{1,m}=\\{u\in C(\bar{D}_{m},\mathbb{R})\\},$

and

$X_{1,m}^{r}=\mathcal{D}(A_{1,m}^{\alpha}),$

where $A_{1,m}$ is $-\Delta$ with Dirichlet boundary condition acting on
$X_{1,m}\cap C_{0}(D_{m})$ and $0<\alpha<1$. We denote the solution of (3.27)
by $u_{m}(t,\cdot;s,u_{0})=(\Phi_{m}(t,s)u_{0})(\cdot)$ with
$u(s,\cdot;s,u_{0})=u_{0}(\cdot)\in X_{1,m}^{r}$. By Proposition 3.2, we have

$r(\Phi_{m}(T,0))=e^{\lambda_{1,m}^{r}T},$

where $\lambda^{r}_{1,m}$ is the principal eigenvalue of the following
eigenvalue problem,

$\begin{cases}-\partial_{t}u+\Delta u+a(t,x)u=\lambda u,\quad&x\in D_{m},\cr
u(t+T,x)=u(t,x),\quad&x\in D_{m},\cr u(t,x)=0,\quad&x\in\partial
D_{m}.\end{cases}$

By the dependence of the principle eigenvalue on the domain perturbation (see
[15]), for any $\epsilon>0$, there exists $m_{1}$ such that

$\lambda_{1,m_{1}}^{r}\leq\lambda_{1}^{r}+\frac{\epsilon}{2}.$ (3.28)

Secondly, let $u_{m_{1}}^{r}(\cdot)$ be a positive eigenfunction of
$\Phi_{m_{1}}(T,0)$ corresponding to $r(\Phi_{m_{1}}(T,0))$. By regularity for
parabolic equations, $u_{m_{1}}^{r}\in C^{3}(\bar{D}_{m_{1}})$. Let
$(\Phi_{m_{1}}^{\delta}(t,0)u^{r}_{m_{1}})(x)$ be the solution to

$\begin{cases}u_{t}=\nu_{\delta}\left[\int_{D_{m_{1}}}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+a(t,x)u(t,x),\quad&x\in\bar{D}_{m_{1}},\\\
u(0,x)=u_{m_{1}}^{r}(x).\end{cases}$ (3.29)

Then by Theorem A,

$\big{(}\Phi_{m_{1}}^{\delta}(nT,0)u_{m_{1}}^{r}\big{)}(x)\leq\big{(}\Phi_{m_{1}}(nT,0)u_{m_{1}}^{r}\big{)}(x)+C(nT,\delta)\quad\forall\,\,x\in\bar{D}_{m_{1}},$

where $C(nT,\delta)\to 0$ as $\delta\to 0$. By Proposition 2.1,

$\big{(}\Phi^{\delta}(nT,0)u_{m_{1}}^{r}|_{\bar{D}}\big{)}(x)\leq\big{(}\Phi_{m_{1}}^{\delta}(nT,0)u_{m_{1}}^{r}\big{)}(x)\quad\forall\,\,x\in\bar{D}.$

It then follows that for $x\in\bar{D}$,

$\displaystyle\big{(}\Phi^{\delta}(nT,0)u_{m_{1}}^{r}|_{\bar{D}}\big{)}(x)$
$\displaystyle\leq\big{(}\Phi_{m_{1}}(nT,0)u_{m_{1}}^{r}\big{)}(x)+C(nT,\delta)$
$\displaystyle=e^{\lambda_{m_{1}}^{r}nT}u_{m_{1}}^{r}(x)+C(nT,\delta)$
$\displaystyle\leq
e^{(\lambda_{1}^{r}+\frac{\epsilon}{2})nT}u_{m_{1}}^{r}(x)+C(nT,\delta)$
$\displaystyle=e^{(\lambda_{1}^{r}+\epsilon)nT}e^{-\frac{\epsilon}{2}nT}u_{m_{1}}^{r}(x)+C(nT,\delta).$
(3.30)

Note that

$\min_{x\in\bar{D}}u_{m_{1}}^{r}(x)>0.$

Hence for any $\epsilon>0$, there is $n_{2}\geq 1$ such that

$e^{-\frac{\epsilon}{2}n_{2}T}\leq\frac{1}{2},$ (3.31)

and there is $\delta_{2}>0$ such that for $0<\delta<\delta_{2}$,

$C(n_{2}T,\delta)\leq\frac{1}{2}e^{(\lambda_{1}^{r}+\epsilon)n_{2}T}u_{m_{1}}^{r}(x)\quad\forall\,x\in\bar{D}.$
(3.32)

By (3.2)-(3.32),

$\big{(}\Phi^{\delta}(n_{2}T,0)u_{m_{1}}^{r}|_{\bar{D}}\big{)}(x)\leq
e^{(\lambda_{1}^{r}+\epsilon)n_{2}T}u_{m_{1}}^{r}(x)\quad\forall\,\,x\in\bar{D}.$

This together with Proposition 2.1 implies that for $0<\delta<\delta_{2}$,

$\big{(}\Phi^{\delta}(nn_{2}T,0)u_{m_{1}}^{r}|_{\bar{D}}\big{)}(x)\leq
e^{(\lambda_{1}^{r}+\epsilon)nn_{2}T}u_{m_{1}}^{r}(x)\quad\forall\,\,x\in\bar{D}.$
(3.33)

This together with Proposition 3.1 implies that

$\lambda_{1}^{\delta}\leq\lambda_{1}^{r}+\epsilon$

for $0<\delta<\delta_{2}$, that is, (3.26) holds.

Theorem B in the Dirichlet boundary condition case then follows from (3.17)
and (3.26). ∎

###  3.3 Proofs of Theorem B in the Neumann and periodic boundary condition
cases

###### Proof of Theorem B in the Neumann boundary condition case.

We assume $B_{r,b}u=B_{r,N}u$ in (1.10), and $D_{b}=D_{N}(=\emptyset)$ in
(1.11). The proof in the Neumann boundary condition case is similar to the
arguments in the Dirichlet boundary condition case (it is simpler). For the
completeness, we give a proof in the following. Without loss of generality, we
may also assume that $a\in\mathcal{X}_{2}\cap
C^{3}(\mathbb{R}\times\mathbb{R}^{N})$.

For the simplicity in notation, put

$\Phi(nT,0)=\Phi_{2}(nT,0;a),\quad\,\ \lambda_{2}^{r}=\lambda_{2}^{r}(a),$

and

$\Phi^{\delta}(nT,0)=\Phi_{2}^{\delta}(nT,0;a),\quad\lambda_{2}^{\delta}=\lambda_{2}^{\delta}(a).$

By Propositions 3.1 and 3.2,

$r(\Phi(T,0))=e^{\lambda_{2}^{r}T},$ (3.34)

and

$r(\Phi^{\delta}(T,0))=e^{\lambda_{2}^{\delta}T}.$ (3.35)

Let $u^{r}(\cdot)$ be a positive eigenfunction of $\Phi(T,0)$ corresponding to
$r(\Phi(T,0))$. By regularity for parabolic equations, $u^{r}\in
C^{3}(\bar{D})$. By Theorem A, we have

$\|\Phi^{\delta}(nT,0)u^{r}-\Phi(nT,0)u^{r}\|_{X_{2}}\leq C(nT,\delta),$

where $C(nT,\delta)\to 0$ as $\delta\to 0$. This implies that for all
$x\in\bar{D}$,

$\displaystyle\big{(}\Phi^{\delta}(nT,0)u^{r}\big{)}(x)$
$\displaystyle\geq\big{(}\Phi(nT,0)u^{r}\big{)}(x)-C(nT,\delta)$
$\displaystyle=e^{\lambda_{2}^{r}nT}u^{r}(x)-C(nT,\delta)$
$\displaystyle=e^{(\lambda_{2}^{r}-\epsilon)nT}e^{\epsilon
nT}u^{r}(x)-C(nT,\delta),$ (3.36)

and

$\displaystyle\big{(}\Phi^{\delta}(nT,0)u^{r}\big{)}(x)$
$\displaystyle\leq\big{(}\Phi(nT,0)u^{r}\big{)}(x)+C(nT,\delta)$
$\displaystyle=e^{\lambda_{2}^{r}nT}u^{r}(x)+C(nT,\delta)$
$\displaystyle=e^{(\lambda_{2}^{r}+\epsilon)nT}e^{-\epsilon
nT}u^{r}(x)+C(nT,\delta).$ (3.37)

Note that

$\min_{x\in\bar{D}}u^{r}(x)>0.$ (3.38)

Hence for any $\epsilon>0$, there is $n_{1}>1$ such that

$\begin{cases}e^{\epsilon
n_{1}T}u^{r}(x)\geq\frac{3}{2}u^{r}(x)\quad\forall\,x\in\bar{D},\\\ \\\
e^{-\epsilon
n_{1}T}u^{r}(x)\leq\frac{1}{2}u^{r}(x)\quad\forall\,x\in\bar{D},\end{cases}$
(3.39)

and there is $\delta_{0}>0$ such that for any $0<\delta<\delta_{0}$,

$C(n_{1}T)\delta<\frac{1}{2}e^{(\lambda_{2}^{r}-\epsilon)n_{1}T}u^{r}(x)\quad\forall\,x\in\bar{D}.$
(3.40)

By (3.3)-(3.40), we have that for any $0<\delta<\delta_{0}$,

$e^{(\lambda_{2}^{r}-\epsilon)n_{1}T}u^{r}(x)\leq\big{(}\Phi^{\delta}(n_{1}T,0)u^{r}\big{)}(x)\leq
e^{(\lambda_{2}^{r}+\epsilon)n_{1}T}u^{r}(x)\quad\forall\,x\in\bar{D}.$

This together with Proposition 2.1 implies that for all $n\geq 1$,

$e^{(\lambda_{2}^{r}-\epsilon)n_{1}nT}u^{r}(x)\leq\big{(}\Phi^{\delta}(n_{1}nT,0)u^{r}\big{)}(x)\leq
e^{(\lambda_{2}^{r}+\epsilon)n_{1}nT}u^{r}(x)\quad\forall\,x\in\bar{D}.$

It then follows that for any $0<\delta<\delta_{0}$,

$e^{(\lambda_{2}^{r}-\epsilon)T}\leq r(\Phi^{\delta}(T,0))\leq
e^{(\lambda_{2}^{r}+\epsilon)T}.$

By Proposition 3.1, we have

$|\lambda_{2}^{\delta}-\lambda_{2}^{r}|<\epsilon\quad\forall\,0<\delta<\delta_{0}.$

Theorem B in the Neumann boundary condition case is thus proved. ∎

###### Proof of Theorem B in the periodic boundary condition case.

We assume $D=\mathbb{R}^{N}$, and $B_{r,b}u=B_{r,P}u$ in (1.10), and
$B_{n,b}u=B_{n,P}u$ in (1.11). It can be proved by the same arguments as in
the Neumann boundary condition case. ∎

##  4 Approximation of Time Periodic Positive Solutions of Random Dispersal
KPP Equations by Nonlocal Dispersal KPP Equations

In this section, we study the approximation of the asymptotic dynamics of time
periodic KPP equations with random dispersal by those of time periodic KPP
equations with nonlocal dispersal. We first recall the existing results about
time periodic positive solutions of KPP equations with random as well as
nonlocal dispersal. Then we prove Theorem C. Throughout this section, we
assume that $D$ is as in (H0), and (H1), (H2) and (H2)δ hold. Recall that,
(H2) implies (H2)δ for $\delta$ sufficiently small by Theorem B.

###  4.1 Basic properties

In this subsection, we present some basic known results for (1.12) and (1.13).
Let $X_{1}^{r}$, $X_{2}^{r}$, and $X_{3}^{r}$ be defined as in (3.7), (3.8),
and (3.9), respectively. For $u_{0}\in X_{i}^{r}$, let
$(U(t,0)u_{0})(\cdot)=u(t,\cdot;u_{0})$, where $u(t,\cdot;u_{0})$ is the
solution to (1.12) with $u(0,\cdot;u_{0})=u_{0}(\cdot)$ and
$B_{r,b}u=B_{r,D}u$ when $i=1$, $B_{r,b}u=B_{r,N}u$ when $i=2$, and
$B_{r,b}u=B_{r,P}u$ when $i=3$. Similarly, for $u_{0}\in X_{i}$, let
$(U^{\delta}(t,0)u_{0})(\cdot)=u^{\delta}(t,\cdot;u_{0})$, where
$u^{\delta}(t,\cdot;u_{0})$ is the solution to (1.13) with
$u^{\delta}(0,\cdot;u_{0})=u_{0}(\cdot)$ and
$D_{b}=D_{D}(=\mathbb{R}^{N}\backslash\bar{D})$, $B_{n,b}u=B_{n,D}u$ when
$i=1$, $D_{b}=D_{N}(=\emptyset)$ when $i=2$, and $B_{n,b}u=B_{n,P}u$ and
$D_{b}=D_{p}(=\mathbb{R}^{N})$ when $i=3$.

######  Proposition 4.1.

  * (1)

If $u_{0}\geq 0$, solution $u(t,\cdot;u_{0})$ to (1.12) with
$u(0,\cdot;u_{0})=u_{0}(\cdot)$ exists for all $t\geq 0$ and
$u(t,\cdot;u_{0})\geq 0$ for all $t\geq 0$.

  * (2)

If $u_{0}\geq 0$, solution $u(t,\cdot;u_{0})$ to (1.13) with
$u(0,\cdot;u_{0})=u_{0}(\cdot)$ exists for all $t\geq 0$ and
$u(t,\cdot;u_{0})\geq 0$ for all $t\geq 0$.

###### Proof.

(1) Note that $u(\cdot)\equiv 0$ is a sub-solution of (1.12) and
$u(\cdot)\equiv M$ is a super-solution of (1.12) for $M\gg 1$. Then by
Proposition 2.1, there is $M\gg 1$ such that

$0\leq u(t,x;u_{0})\leq M\quad\forall\,\,x\in\bar{D},\,\,t\in(0,t_{\max}),$

where $(0,t_{\max})$ is the interval of existence of $u(t,\cdot;u_{0})$. This
implies that we must have $t_{\max}=\infty$ and hence (1) holds.

(2) It can be proved by similar arguments as in (1). ∎

######  Proposition 4.2.

  * (1)

(1.12) has a unique globally stable positive time periodic solution
$u^{*}(t,x)$.

  * (2)

(1.13) has a unique globally stable time periodic positive solution
$u^{*}_{\delta}(t,x)$.

###### Proof.

(1) See [45, Theorem 3.1] (see also [33, Theorems 1.1, 1.3]).

(2) See [36, Theorem E]. ∎

######  Remark 4.1.

By Proposition 4.2(2), if there is $u_{0}\in X_{i}^{+}\setminus\\{0\\}$ such
that $(U^{\delta}(nT,0)u_{0})(\cdot)\geq u_{0}(\cdot)$ for some $n\geq 1$,
then we must have
$\lim_{n\to\infty}(U^{\delta}(nT,0)u_{0})(\cdot)=u^{*}_{\delta}(0,\cdot)$ and
hence

$(U^{\delta}(nT,0)u_{0})(\cdot)\leq u^{*}_{\delta}(0,\cdot).$

Similarly, if there is $u_{0}\in X_{i}^{+}\setminus\\{0\\}$ such that
$(U^{\delta}(nT,0)u_{0})(\cdot)\leq u_{0}(\cdot)$ for some $n\geq 1$, then

$(U^{\delta}(nT,0)u_{0})(\cdot)\geq u^{*}_{\delta}(0,\cdot).$

###  4.2 Proof of Theorem C in the Dirichlet boundary condition case

In this subsection, we prove Theorem C in the Dirichlet boundary condition
case. Throughout this subsection, we assume that $B_{r,b}u=B_{r,D}u$ in
(1.12), and $D_{b}=D_{D}$ and $B_{n,b}u=B_{n,D}u$ in (1.13).

###### Proof of Theorem C in the Dirichlet boundary condition case.

First of all, note that it suffices to prove that for any $\epsilon>0$, there
is $\delta_{0}>0$ such that for $0<\delta<\delta_{0}$,

$u_{\delta}^{*}(t,x)-\epsilon\leq u^{*}(t,x)\leq
u_{\delta}^{*}(t,x)+\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$

We first show that for any $\epsilon>0$, there is $\delta_{1}>0$ such that for
$0<\delta<\delta_{1}$,

$u^{*}(t,x)\leq
u_{\delta}^{*}(t,x)+\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.1)

To this end, choose a smooth function $\phi_{0}\in C_{0}^{\infty}(D)$
satisfying that $\phi_{0}(x)\geq 0$ for $x\in D$ and
$\phi_{0}(\cdot)\not\equiv 0$. Let $0<\eta\ll 1$ be such that

$u_{-}(x):=\eta\phi_{0}(x)<u^{*}(0,x)\quad{\rm for}\quad x\in\bar{D}.$

Then there is $\epsilon_{0}>0$ such that

$u^{*}(0,x)\geq u_{-}(x)+\epsilon_{0}\quad{\rm for}\quad x\in{\rm
supp}(\phi_{0}).$ (4.2)

By Proposition 4.2, there is $N\gg 1$ such that

$\big{(}U(NT,0)u_{-}\big{)}(x)\geq
u^{*}(NT,x)-\epsilon_{0}/2=u^{*}(0,x)-\epsilon_{0}/2\quad\forall\,\,x\in\bar{D}.$

By Theorem A, there is $\bar{\delta}_{1}>0$ such that for
$0<\delta<\bar{\delta}_{1}$, we have

$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq\big{(}U(NT,0)u_{-}\big{)}(x)-\epsilon_{0}/2\quad\forall\,\,x\in\bar{D}.$

Hence for $0<\delta<\bar{\delta}_{1}$,

$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq
u^{*}(0,x)-\epsilon_{0}\quad\forall\,\,x\in\bar{D}.$ (4.3)

Note that

$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq 0\quad\forall\,\,x\in\bar{D}.$

It then follows from (4.2) and (4.3) that for $0<\delta<\bar{\delta}_{1}$,

$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq
u_{-}(x)\quad\forall\,\,x\in\bar{D}.$

This together with Proposition 4.2 (2) implies that

$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\leq
u_{\delta}^{*}(0,x)\quad\forall\,\,x\in\bar{D}$ (4.4)

(see Remark 4.1).

By Proposition 4.2 (1) again, for $n\gg 1$,

$u^{*}(t,x)\leq(U(nNT+t,0)u_{-})(x)+\epsilon/2\,\,\,\,\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.5)

Fix an $n\gg 1$ such that (4.5) holds. By Theorem A, there is
$0<\tilde{\delta}_{1}\leq\bar{\delta}_{1}$ such that for
$0<\delta<\tilde{\delta}_{1}$,

$\displaystyle(U(nNT+t,0)u_{-})(x)\leq(U^{\delta}(nNT+t,0)u_{-})(x)+C_{1}(\delta),$
(4.6)

where $C_{1}(\delta)\to 0$ as $\delta\to 0$. By (4.4), Proposition 2.1, and
Proposition 4.2 (2),

$\big{(}U^{\delta}(nNT+t,0)u_{-}\big{)}(x)\leq\big{(}U^{\delta}(t,0)u^{*}_{\delta}(0,\cdot)\big{)}(x)=u_{\delta}^{*}(t,x)$
(4.7)

for $t\in[0,T]$ and $x\in\bar{D}$. Let $0<\delta_{1}\leq\tilde{\delta}_{1}$ be
such that

$C_{1}(\delta)<\epsilon/2\quad\forall\,\,0<\delta<\delta_{1}.$ (4.8)

(4.1) then follows from (4.5)-(4.8).

Next, we need to show for any $\epsilon>0$, there is $\delta_{2}>0$ such that
for $0<\delta<\delta_{2}$,

$u^{*}(t,x)\geq
u_{\delta}^{*}(t,x)-\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.9)

To this end, choose a sequence of open sets $\\{D_{m}\\}$ with smooth
boundaries such that $D_{1}\supset D_{2}\supset D_{3}\cdots\supset
D_{m}\supset\cdots\supset\bar{D}$, and $\bar{D}=\cap_{m=1}^{\infty}D_{m}$.
According to Corollary 5.11 in [17], $D_{m}\to D$ regularly and all assertions
of Theorem 5.5 in [17] hold.

Consider

$\begin{cases}\partial_{t}u=\Delta u+uf(t,x,u),\quad&x\in D_{m},\\\
u(t,x)=0,\quad&x\in\partial D_{m}.\end{cases}$ (4.10)

Let $U_{m}(t,0)u_{0}=u(t,\cdot;u_{0})$, where $u(t,\cdot;u_{0})$ is the
solution to (4.10) with $u(0,\cdot;u_{0})=u_{0}(\cdot)$. By Proposition 4.2,
(4.10) has a unique time periodic positive solution $u_{m}^{*}(t,x)$. We first
claim that

$\lim_{m\to\infty}u_{m}^{*}(t,x)\to u^{*}(t,x)\,\,\,\text{uniformly
in}\,\,t\in[0,T]\,\,\text{and}\,\,x\in\bar{D}.$ (4.11)

In fact, it is clear that $u^{*}\in C(\mathbb{R}\times\bar{D},\mathbb{R})$ and
$u_{m}^{*}\in C(\mathbb{R}\times\bar{D}_{m},\mathbb{R})$. By [15, Theorem
7.1],

$\sup_{t\in\mathbb{R}}\|u_{m}^{*}(t,\cdot)-u^{*}(t,\cdot)\|_{L^{q}(D)}\to
0\quad{\rm as}\quad m\to\infty$

for $1\leq q<\infty$. Let $a(t,x)=f(t,x,u^{*}(t,x))$ and
$a_{m}(t,x)=f(t,x,u_{m}^{*}(t,x))$. Then $u^{*}(t,x)$ and $u_{m}^{*}(t,x)$ are
time periodic solutions to the following linear parabolic equations,

$\begin{cases}u_{t}=\Delta u+a(t,x)u,\quad&x\in D,\cr u(t,x)=0,&x\in\partial
D,\end{cases}$ (4.12)

and

$\begin{cases}u_{t}=\Delta u+a_{m}(t,x)u,\quad&x\in D_{m},\cr
u(t,x)=0,&x\in\partial D_{m},\end{cases}$ (4.13)

respectively. Observe that there is $M>0$, such that

$\|a\|_{L^{\infty}([T,2T]\times D)}<M,\,\,\|a_{m}\|_{L^{\infty}([T,2T]\times
D_{m})}<M,\,\,\|u^{*}(0,\cdot)\|_{L^{\infty}(D)}<M,\text{ and
}\|u_{m}^{*}(0,\cdot)\|_{L^{\infty}(D_{m})}<M.$

By [1, Theorem D(1)], $\\{u_{m}^{*}(t,x)\\}$ is equi-continuous on
$[T,2T]\times\bar{D}$. Without loss of generality, we may then assume that
$u_{m}^{*}(t,x)$ converges uniformly on $[T,2T]\times\bar{D}$. But
$u_{m}^{*}(t,\cdot)\to u^{*}(t,\cdot)$ in $L^{q}(D)$ uniformly in $t$. We then
must have

$u_{m}^{*}(t,x)\to u^{*}(t,x)\quad{\rm as}\quad n\to\infty$

uniformly in $(t,x)\in[T,2T]\times\bar{D}$. This together with the time
periodicity of $u_{m}^{*}$ shows that (4.11) holds.

Next, for any $\epsilon>0$, fix $m\gg 1$ such that

$u^{*}(t,x)\geq
u_{m}^{*}(t,x)-\epsilon/3\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.14)

Choose $M\gg 1$ such that for $0<\delta\leq 1$,

$Mu_{m}^{*}(t,x)\geq
u_{\delta}^{*}(t,x)\,\,\,\,\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.15)

Let

$u_{m}^{+}(x)=Mu_{m}^{*}(0,x),\quad u^{+}(x)=u_{m}^{+}(x)|_{\bar{D}}.$

By Proposition 4.2, for fixed $m$ and $\epsilon$, there exists $N\gg 1$, such
that

$u_{m}^{*}(t,x)\geq\big{(}U_{m}(NT+t,0)u_{m}^{+}\big{)}(x)-\epsilon/3\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.16)

By Theorem A, there is $0<\tilde{\delta}_{2}<1$ such that for
$0<\delta<\tilde{\delta}_{2}$,

$(U_{m}(NT+t,0)u_{m}^{+})(x)\geq(U_{m}^{\delta}(NT+t,0)u_{m}^{+})(x)-C_{2}(\delta)\quad\forall\,t\in[0,T],\,\
x\in D_{m},$ (4.17)

where $C_{2}(\delta)\to 0$ as $\delta\to 0$ and
$(U^{\delta}_{m}(t,0)u_{0})(\cdot)=u(t,\cdot;u_{0})$ is the solution to

$\begin{cases}u_{t}(t,x)=\nu_{\delta}\left[\int_{D_{m}}k_{\delta}(y-x)u(t,y)dy-u(t,x)\right]+u(t,x)f(t,x,u(t,x)),\quad&x\in\bar{D}_{m}\cr
u(0,x)=u_{0}(x),&x\in\bar{D}_{m}.\end{cases}$

Let $0<\delta_{2}<\tilde{\delta}_{2}$ be such that for $0<\delta<\delta_{2}$,

$C_{2}(\delta)<\epsilon/3.$ (4.18)

By Proposition 2.1, for $x\in\bar{D}$ we have

$(U_{m}^{\delta}(NT+t,0)u_{m}^{+})(x)\geq(U^{\delta}(NT+t,0)u^{+})(x),$

and

$(U^{\delta}(NT+t,0)u^{+})(x)=(U^{\delta}(t,0)U^{\delta}(NT,0)u^{+})(x)\geq(U^{\delta}(t,0)u_{\delta}^{*}(0,\cdot))(x)=u_{\delta}^{*}(t,x).$

This together with (4.14), (4.16), (4.17), and (4.18) implies (4.9).

So, for any $\epsilon>0$, there exists
$\delta_{0}=\min\\{\delta_{1},\delta_{2}\\}$, such that for any
$\delta<\delta_{0}$, we have

$|u^{*}(t,x)-u_{\delta}^{*}(t,x)|\leq\epsilon\,\ \text{ uniform in }t>0\text{
and }x\in\bar{D}.$

∎

###  4.3 Proofs of Theorem C in the Neumann and periodic boundary condition
cases

In this subsection, we prove Theorem C in the Neumann and periodic boundary
condition cases.

###### Proof of Theorem C in the Neumann boundary condition case.

We assume $B_{r,b}u=B_{r,N}u$ in (1.10), and $D_{b}=D_{N}(=\emptyset)$ in
(1.11). The proof in the Neumann boundary condition case is similar to the
arguments in the Dirichlet boundary condition case (it is indeed simpler). For
completeness, we provide a proof.

First, we show that for any $\epsilon>0$, there is $\delta_{1}>0$ such that

$u^{*}(t,x)\leq
u_{\delta}^{*}(t,x)+\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D},$ (4.19)

if $0<\delta<\delta_{1}$. Choose a smooth function $u_{-}\in
C^{\infty}(\bar{D})$ with $u_{-}(\cdot)\geq 0$ and $u_{-}(\cdot)\not\equiv 0$
such that

$u_{-}(x)<u^{*}(0,x)\quad\forall\,\,x\in\bar{D}.$

Then there is $\epsilon_{0}>0$ such that

$u^{*}(0,x)\geq u_{-}(x)+\epsilon_{0}\quad\forall\,\,x\in\bar{D}.$ (4.20)

By Proposition 4.2 (1), there is $N\gg 1$ such that

$\big{(}U(NT,0)u_{-}\big{)}(x)\geq
u^{*}(0,x)-\epsilon_{0}/2\quad\forall\,\,x\in\bar{D}.$ (4.21)

By Theorem A, there is $\bar{\delta}_{1}>0$ such that for
$0<\delta<\bar{\delta}_{1}$,

$(U^{\delta}(NT,0)u_{-})(x)\geq(U(NT,0)u_{-})(x)-\epsilon_{0}/2\quad\forall\,\,x\in\bar{D}.$
(4.22)

By (4.20), (4.21) and (4.22),

$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\geq
u_{-}(x)\quad\forall\,\,x\in\bar{D},$

and then by Proposition 4.2 (2),

$\big{(}U^{\delta}(NT,0)u_{-}\big{)}(x)\leq
u_{\delta}^{*}(0,x)\quad\forall\,\,x\in\bar{D}.$ (4.23)

By Proposition 4.2 (1) again, for any given $\epsilon>0$, $n\gg 1$, and
$0<\delta<\bar{\delta}_{1}$,

$\displaystyle u^{*}(t,x)$
$\displaystyle\leq(U(nNT+t,0)u_{-})(x)+\epsilon/2\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.24)

By Theorem A, there is $0<\delta_{1}\leq\bar{\delta}_{1}$ such that for
$\delta<\delta_{1}$,

$\displaystyle(U(nNT+t,0)u_{-})(x)\leq(U^{\delta}(nNT+t,0)u_{-})(x)+\frac{\epsilon}{2}\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.25)

By Proposition 2.1 and (4.23), we have

$(U^{\delta}(nNT+t,0)u_{-})(x)=(U^{\delta}(t,0)U^{\delta}(nNT,0)u_{-})(x)\leq(U^{\delta}(t,0)u^{*}_{\delta}(t,\cdot))(x)=u_{\delta}^{*}(t,x)$
(4.26)

for $t\in[0,T]$ and $x\in\bar{D}$. (4.19) then follows from (4.24)-(4.26).

Next, we show that for any $\epsilon>0$, there is $\delta_{2}>0$ such that for
$0<\delta<\delta_{2}$,

$u^{*}(t,x)\geq
u_{\delta}^{*}(t,x)-\epsilon\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$ (4.27)

Choose $M\gg 1$ such that $f(t,x,M)<0$ for $t\in\mathbb{R}$ and $x\in\bar{D}$.
Put

$u^{+}(x)=M\quad\forall\,\,x\in\bar{D}.$

Then for all $\delta>0$,

$u_{\delta}^{*}(0,x)\leq u^{+}(x)\quad\forall\,\,x\in\bar{D}.$ (4.28)

By Proposition 4.2, there is $N\gg 1$ such that

$u^{*}(t,x)\geq(U(NT+t,0)u^{+})(x)-\epsilon/2\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.29)

By Theorem A, there is $\delta_{2}>0$ such that for $0<\delta<\delta_{2}$,

$(U(NT+t,0)u^{+})(x)\geq(U^{\delta}(NT+t,0)u^{+})(x)-\frac{\epsilon}{2}\quad\forall\,\,t\in[0,T],\,\,x\in\bar{D}.$
(4.30)

By (4.28),

$(U^{\delta}(NT+t,0)u^{+})(x)=(U^{\delta}(t,0)U^{\delta}(NT,0)u^{+})(x)\geq(U^{\delta}(t,0)u_{\delta}^{*}(t,\cdot))(x)=u_{\delta}^{*}(t,x)$
(4.31)

for $t\in[0,T]$ and $x\in\bar{D}$. (4.27) then follows from (4.29)-(4.31).

So, for any $\epsilon>0$, there exists
$\delta_{0}=\min\\{\delta_{1},\delta_{2}\\}$, such that for any
$0<\delta<\delta_{0}$, we have

$|u^{*}(t,x)-u_{\delta}^{*}(t,x)|\leq\epsilon\,\ \text{ uniform in }t>0\text{
and }x\in\bar{D}.$

∎

###### Proof of Theorem C in the periodic boundary condition case.

We assume $D=\mathbb{R}^{N}$, and $B_{r,b}u=B_{r,P}u$ in (1.10), and
$B_{n,b}u=B_{n,P}u$ in (1.11). It can be proved by the similar arguments as in
the Neumann boundary condition case. ∎

Acknowledgments. The authors would like to thank the referees for the valuable
comments and suggestions which improved the presentation of this paper
considerably.

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