# On a cross-diffusive SIS epidemic model with power-like nonlinear incidence

Huicong Li  School of Mathematics, Sun Yat-sen University, Guangzhou, 510275,
Guangdong Province, China<EMAIL_ADDRESS>and  Tian Xiang†
Institute for Mathematical Sciences, Renmin University of China, Bejing,
100872, China<EMAIL_ADDRESS>
###### Abstract.

We study global existence, boundedness and convergence of nonnegative
classical solutions of a Neumann initial-boundary value problem for the
following cross-diffusive SIS (susceptible-infected-susceptible) epidemic
model with power-like infection mechanism generalizing the standard mass
action mechanism:

$\begin{cases}S_{t}=d_{S}\Delta S+\chi\nabla\cdot(S\nabla I)-\beta
S^{q}I^{p}+\gamma I,&x\in\Omega,\,t>0,\\\\[7.11317pt] I_{t}=d_{I}\Delta
I+\beta S^{q}I^{p}-(\gamma+\mu)I,&x\in\Omega,\,t>0,\end{cases}$

in a bounded smooth domain $\Omega\subset\mathbb{R}^{n}\ (n\geq 1)$. The
incidence of the form $\beta S^{q}I^{p}$ with $p,\,q>0$ is a natural extension
of the classical mass action type $\beta SI$, and the cross-diffusive term
$\chi\nabla\cdot(S\nabla I)$ with $\chi\geq 0$ describes the effect that the
susceptible individuals tend to move away from higher concentration of
infected populations. Global existence and boundedness of classical solutions
are established in certain parameter ranges, and threshold/non-threshold long-
time behaviors of global bounded solutions are also detected. Our findings
significantly improve and extend previous related studies.

###### Key words and phrases:

epidemic models, cross-diffusion, global existence and boundedness, threshold
dynamics, non-threshold dynamics.

###### 2010 Mathematics Subject Classification:

35K57, 35Q92, 35A01, 35B40, 92D30

† Corresponding author.

##  1\. Introduction

In the theory of epidemiology, incidence is the occurrence of new cases of a
disease via contact between infected individuals and susceptible ones within a
specified period of time, and it is usually characterized by the infection
mechanism in the modeling of infectious diseases. In the seminal work of
Kermack and McKendrick [19], the authors studied a susceptible-infected-
recovered compartmental model with bilinear incidence rate $\beta SI$, where
$S$ and $I$, respectively, are the numbers of susceptible and infected
populations, and the positive constant $\beta$ measures the disease
transmission rate. However, as argued in [12, 35], this type of bilinear
infection mechanism, describing the simple mass action law, has its own
shortcomings and may not fit the field data well. Thus, various different
types of transmission mechanism were proposed and investigated, such as the
general nonlinear transmission rate of the form $\beta S^{q}I^{p}$ with $p$,
$q>0$ [15, 20, 23, 24, 30, 32, 33, 40, 44], the standard incidence type of
$\beta\frac{SI}{S+I}$ [1, 2, 9, 10, 11, 21, 25, 26, 27, 38, 39, 41, 42], the
saturating incidence rate $\beta\frac{SI}{1+mI}$ with parameter $m>0$ [6, 8,
17, 31, 34, 43, 51, 63], etc; see also references therein for many other types
of nonlinear infection mechanisms.

In this paper, inspired mainly from [13, 29, 40], we are interested in the
dynamical properties of solutions to the following homogeneous Neumann IBVP
(initial-boundary value problem) of (possibly) cross-diffusive SIS epidemic
model with generalized mass action infection mechanism:

$\begin{cases}S_{t}=d_{S}\Delta S+\chi\nabla\cdot(S\nabla I)-\beta
S^{q}I^{p}+\gamma I,&x\in\Omega,\,t>0,\\\\[7.11317pt] I_{t}=d_{I}\Delta
I+\beta S^{q}I^{p}-(\gamma+\mu)I,&x\in\Omega,\,t>0,\\\\[7.11317pt]
\frac{\partial S}{\partial\nu}=\frac{\partial
I}{\partial\nu}=0,&x\in\partial\Omega,\,t>0,\\\\[7.11317pt] S(x,0)=S_{0}(x),\
\ I(x,0)=I_{0}(x),&x\in\Omega.\end{cases}$ (1.1)

Here and below, $\Omega\subset\mathbb{R}^{n}~{}(n\geq 1)$ is a bounded domain
with smooth boundary $\partial\Omega$. The unknown functions $S=S(x,t)$ and
$I=I(x,t)$, respectively, denote the population density of susceptible and
infected individuals at location $x$ and time $t$; $d_{S}$ and $d_{I}$ are
positive constants measuring the random mobility of susceptible and infected
populations respectively; the featured cross-diffusion term
$\chi\nabla\cdot(S\nabla I)$ describes the biased movement (widely known as
repulsive chemotaxis effect in chemotactic models) of those “smart”
susceptible individuals who tend to stay away from regions of higher
concentration of infected individuals with nonnegative constant $\chi$
measuring the strength of such effect; positive constants $\beta$ and
$\gamma$, respectively, stand for the disease transmission rate and recovery
rate, whereas the parameter $\mu\geq 0$ is the disease-induced death rate. The
homogeneous Neumann boundary conditions imply there is no population flux
across the boundary $\partial\Omega$. As explained in [40], we assume within
this paper that the initial datum $(S_{0},I_{0})$ satisfies

$\begin{cases}S_{0}\geq 0,\ I_{0}\geq,\,\not\equiv 0,\ (S_{0},I_{0})\in
C^{0}(\overline{\Omega})\times W^{1,r_{0}}(\Omega)\text{ for some
}r_{0}>\max\\{2,n\\},\\\\[5.69046pt] \inf_{\Omega}S_{0}>0,\ \ \text{ if
}0<q<1,\\\\[5.69046pt] \inf_{\Omega}I_{0}>0,\ \ \text{ if }0<p<1.\end{cases}$
(1.2)

The PDE model (1.1) with $\chi=0$, $q=p=1$ and $\mu=0$ was proposed and
studied by Deng and Wu [13], wherein they established the global
attractiveness of the DFE (disease-free equilibrium) and EE (endemic
equilibrium) in some special cases, and the existence (and uniqueness) of EE
in a heterogeneous environment was also discussed when the basic reproduction
number is greater than one. The asymptotic behavior of EE with respect to
diffusion rates $d_{S}$ and $d_{I}$ was investigated later in [60]; see also
further development in [7, 52, 28]. Recently, Peng and Wu [40] studied the
reaction-diffusion model (1.1) without cross-diffusion ($\chi=0$) and they
obtained the global existence and boundedness of classical solutions for any
$p,\,q>0$ by using a duality argument, and then, they derived long time
behaviors of solutions for $\mu>0$ and uniform persistence for $\mu=0$ upon
establishing the ultimately uniform boundedness. More dynamics on many special
cases of the reaction-diffusion model (1.1) without cross-diffusion can be
found in the introduction of [40].

Due to the strong coupling in the (highest) 2nd order term, in study of
chemotaxis systems, it is well-known that the chemotactic cross-diffusion
$\chi\nabla\cdot(S\nabla I)$ has a strong tendency towards driving the
solutions of the underlying models to blow-up in finite or infinite time; see
[3, 54, 56] for example. Compared to the extensive literature on various
chemotactic systems, it seems that epidemic models with cross-diffusive terms
were much less studied. To the best of our knowledge, we are only aware of
very few papers [16, 29, 59] dealing with such mathematical models (see [4,
49, 57], etc, for May-Nowak type cross-diffusive models for virus infection).
Moreover, it should be noted that in these works, the infection mechanism is
taken to be the standard incidence type of the form $\beta\frac{SI}{S+I}$ (in
SIS models) or $\beta\frac{SI}{S+I+R}$ (in SIRS models). From the mathematical
point of view, such frequency-dependent transmission term grows at most
linearly in $I$, and thus one can take full advantage of this property to
derive not only the $L^{\infty}$-bound of $I$ from its $L^{1}$-bound, but also
the $L^{\infty}$-bound of $\nabla I$. As a result, global existence and
boundedness of solutions can be established in [16, 29, 59] in any spatial
dimensions, regardless of the magnitude of $\chi$. Nevertheless, in our cross-
diffusive model (1.1) with power-like nonlinear incidence $\beta S^{q}I^{p}$,
the above strategy is no longer available, and the duality argument used in
[40] also fails because of the presence of the chemotactic cross-diffusion.
Even when $p=q=1$, performing similar yet lengthy computations as done in [47,
55, 61] aimed to cancel out the chemotaxis involving term from (1.1), we find
that

$\begin{split}&\frac{d}{dt}\int_{\Omega}\left[2\beta S(\ln
S-1)+2\chi\left|\nabla I^{\frac{1}{2}}\right|^{2}\right]+8\beta
d_{S}\int_{\Omega}\left|\nabla S^{\frac{1}{2}}\right|^{2}\\\
&\quad+4(\gamma+\mu)\chi\int_{\Omega}\left|\nabla
I^{\frac{1}{2}}\right|^{2}+2d_{I}\chi\int_{\Omega}I\left|D^{2}\ln
I\right|^{2}\\\ &=-2\beta^{2}\int_{\Omega}SI\ln
S+2\beta\gamma\int_{\Omega}I\ln
S+d_{I}\chi\int_{\partial\Omega}\frac{1}{I}\frac{\partial\left|\nabla
I\right|^{2}}{\partial\nu}+\beta\chi\int_{\Omega}\frac{S}{I}\left|\nabla
I\right|^{2}.\end{split}$ (1.3)

Thus, the chemotactic cross-diffusion induced term has been offset but the
last term seems uncontrollable in terms of the dissipative terms on the left-
hand-side of (1.3). Even facing such challenges (strong -no saturation-
chemotactic cross-diffusion in the second order term and strong nonlinear
coupling in the zeroth order term), we shall employ subtle energy estimates
and/or semi-group arguments to bootstrap the easily obtained a priori
$L^{1}$-boundedness to $L^{\infty}$-boundedness of classical solutions to the
IBVP (1.1) under either

$np+\left(n-2\right)^{+}q<n+\min\left\\{n,\ 2\right\\},\ \ \ |\chi|<\chi_{0}\
\text{with}\ \chi_{0}\ \text{defined by}\ \eqref{chi-max},$

or

$q<\frac{1}{n+1}\quad\mbox{and}\quad\
p+(n+1)q<1+\min\left\\{1,\frac{2}{n}\right\\},$ (1.4)

or

$\begin{cases}10q+4p<15,\ \ \ \ q+p<3,&\text{ if }n=1,\\\\[7.11317pt] 3q+p<3,\
\ \ \ q+p<2,&\text{ if }n=2.\end{cases}$ (1.5)

We here underline that, under either (1.4) or (1.5), boundedness of classical
solutions to the IBVP (1.1) is ensured without any smallness assumption on the
initial data; see Theorems 3.3, 3.4 and 3.9 for details.

Going beyond boundedness, the second part of this project is devoted to long-
time behaviors of global bounded solutions to the IBVP (1.1). In the presence
of mortality, i.e., $\mu>0$, it is shown in Theorem 4.1 that the usual
threshold dynamics for epidemic models in terms of a so-called basic
reproduction number do not exist. More specifically, as $t\rightarrow\infty$,
any bounded global solution $(S,I)$ converges uniformly on $\overline{\Omega}$
according to

$\left(S(\cdot,t),I(\cdot,t)\right)\rightarrow\begin{cases}(0,0),&\text{ if
}0<p<1,\\\\[5.69046pt] (S_{\infty},0),&\text{ if }p\geq 1.\end{cases}$

Furthermore,
$S_{\infty}\leq\left(\frac{\gamma+\mu}{\beta}\right)^{\frac{1}{q}}$ if $p=1$
and $(S,I)\rightarrow(S_{\infty},0)$ exponentially if $p>1$. Here,
$S_{\infty}$ is a positive constant and is given implicitly by

$S_{\infty}=|\Omega|^{-1}\left(\int_{\Omega}\left(S_{0}+I_{0}\right)-\mu\int_{0}^{\infty}\int_{\Omega}I\right).$
(1.6)

Comparing with [40, Theorem 2.4], we see that the dynamics of the cross-
diffusive system stay the same as that of the system without cross-diffusion,
indicating the effect of chemotaxis is overbalanced by mortality in respective
of long time behavior. Here, we would like to add that, we use a different
approach to derive such refined convergence by identifying $S_{\infty}$ in
(1.6) and showing that the convergence is exponential when $p>1$.

In the absence of mortality and of cross-diffusion, i.e., $\mu=\chi=0$,
allowing $\beta$ and $\gamma$ to be spatially-dependent, via the basic
reproduction number, threshold dynamics of global solutions to (1.1) with
$p=q=1$ was shown in [13] in two special cases, and only uniform persistence
of global solutions is shown in [40]. In the above scenario, we strengthen the
results in [40]. More specifically, for $0<p<1$, we provide non-threshold
dynamics of global and bounded solutions; that is, we prove the global
attractivity of the unique constant or non-constant EE:

  * (C1)

Assume that $0<p<1$ and $\gamma(x)\equiv r\beta(x)$ on $\overline{\Omega}$ for
some $r>0$. Then, as $t\to\infty$,

$\left(S(\cdot,t),I(\cdot,t)\right)\to(S^{*},I^{*})$

uniformly on $\overline{\Omega}$, where $(S^{*},I^{*})$ is the constant EE
defined by (4.22).

  * (C2)

When $\frac{\gamma}{\beta}\not\equiv\text{const}$, assume that $0<p<1$ and
$d_{S}=d_{I}$. Then, as $t\to\infty$,

$\left(S(\cdot,t),I(\cdot,t)\right)\to(\tilde{S},\tilde{I})$

uniformly on $\overline{\Omega}$, where $(\tilde{S},\tilde{I})$ is the
positive non-constant EE of (1.1) with $\chi=0$.

While in the case of $p=1$, we establish threshold dynamics in terms of the
basic reproduction number $\mathcal{R}_{0}$ defined by (4.35):

  * (C3)

Assume that $p=1$ and $\gamma(x)\equiv r\beta(x)$ on $\overline{\Omega}$ for
some $r>0$. Then, as $t\to\infty$,

$\displaystyle\left(S(\cdot,t),I(\cdot,t)\right)\rightarrow\begin{cases}\left(\frac{N}{|\Omega|},0\right),&\text{if
}\mathcal{R}_{0}\leq 1\Longleftrightarrow\frac{N}{|\Omega|}\leq
r^{\frac{1}{q}},\\\\[7.11317pt]
\left(r^{\frac{1}{q}},\frac{N}{|\Omega|}-r^{\frac{1}{q}}\right),&\text{if
}\mathcal{R}_{0}>1\Longleftrightarrow\frac{N}{|\Omega|}>r^{\frac{1}{q}},\end{cases}$

uniformly on $\overline{\Omega}$, where $N=\int_{\Omega}(S_{0}+I_{0})$ is the
conserved total population.

  * (C4)

When $\frac{\gamma}{\beta}\not\equiv\text{const}$, assume that $p=1$ and
$d_{S}=d_{I}$. Then, as $t\to\infty$,

$\displaystyle\left(S(\cdot,t),I(\cdot,t)\right)\rightarrow\begin{cases}\left(\frac{N}{|\Omega|},0\right),&\text{if
}\mathcal{R}_{0}\leq 1,\\\\[7.11317pt] (\tilde{S},\tilde{I}),&\text{if
}\mathcal{R}_{0}>1,\end{cases}$

uniformly on $\overline{\Omega}$, where $(\tilde{S},\tilde{I})$ is the
positive non-constant EE of (1.1) with $\chi=0$.

See Theorems 4.2, 4.4, 4.5 and 4.6 for more details. Particularly, in the
special case that both $\beta$ and $\gamma$ are positive constants, for all
$0<p\leq 1$ and $q>0$, we provide a complete description of the long-time
behavior of solutions to system (1.1) with $\chi=0$. Here, we would like to
mention that the global asymptotic stability of DFE or EE in general case
seems to be rather challenging, which remains largely open even for much
simpler SIS epidemic models than (1.1). We also mention, besides (C1)-(C4),
many of our other results still hold if $\beta$ and $\gamma$ are spatial-
temporally dependent, as long as they are Hölder continuous and bounded from
below and above by positive constants.

The remainder of this project is organized as follows. In Section 2, we
present some preliminary results and the local well-posedness of our model.
Section 3 is devoted to the global existence and boundedness of solutions
where we prove a couple of results based on different approaches. Finally, in
the last section, we shall investigate the large-time convergence of bounded
and global classical solutions.

For convenience and simplicity, in the sequel, the symbols $C$,
$C_{i}(i=1,2,\cdots)$ or $C_{\epsilon}$ will denote generic positive constants
which are independent of time and may vary line-by-line. The spatial
integration symbol $dx$ will also be omitted when no confusion could arise.

##  2\. Preliminary and local well-posednesss

Let us start with the elementary Young’s inequality with epsilon: Let $r$ and
$s$ be two given positive numbers with $\frac{1}{r}+\frac{1}{s}=1$. Then, for
any $\epsilon>0$,

$ab\leq\epsilon a^{r}+\frac{b^{s}}{(\epsilon
r)^{\frac{s}{r}}},\quad\quad\forall\,a,b\geq 0.$

In several places, we need the well-known Gagliardo-Nirenberg interpolation
inequality, we state it here for the convenience of reference.

######  Lemma 2.1 (Gagliardo-Nirenberg interpolation inequality [14, 36]).

Let $\Omega\subset\mathbb{R}^{n}\ (n\geq 1)$ be a bounded and smooth domain.
Let $l$ and $k$ be integers satisfying $0\leq l<k$, and let $1\leq
s,\,r\leq\infty$, $0<m\leq\infty$ and $\frac{l}{k}\leq a\leq 1$ such that

$\frac{1}{m}-\frac{l}{n}=a\left(\frac{1}{s}-\frac{k}{n}\right)+(1-a)\frac{1}{r}.$

Then, for any $f\in W^{k,s}(\Omega)\cap L^{r}(\Omega)$, there exist a positive
constant $C$ depending only on $\Omega,k,r,s$ and $n$ such that

$\|D^{l}f\|_{L^{m}}\leq
C\left(\|D^{k}f\|_{L^{s}}^{a}\|f\|_{L^{r}}^{1-a}+\|f\|_{L^{r}}\right),$ (2.1)

with the following exception: if $1<s<\infty$ and $k-l-\frac{n}{s}$ is a
nonnegative integer, then $\eqref{G-N-I}$ holds only for $a$ satisfying
$\frac{l}{k}\leq a<1$.

Henceforth, for notational convenience, we write $\|f\|_{L^{p}}$ for the usual
$L^{p}(\Omega)$-norm of $f$.

In our subsequent analysis, we shall apply the commonly used smoothing
$L^{r}$-$L^{s}$ estimates of the Neumann heat semi-group
$\left(e^{td\Delta}\right)_{t\geq 0}$ on a bounded and smooth domain $\Omega$;
see, for instance, [5, 29, 53]. Again, we list them here for direct reference.

######  Lemma 2.2.

For $d>0$, let $\left(e^{td\Delta}\right)_{t\geq 0}$ be the Neumann heat semi-
group and $\lambda_{1}=:\lambda_{1}(d)>0$ be the first positive Neumann
eigenvalue of $-d\Delta$ on $\Omega$. Then there exist some positive constants
$k_{i}$ ($i=1,2,3,4$) depending only on $d$ and $\Omega$ fulfilling the
following smoothing $L^{r}$-$L^{s}$ type estimates:

  1. (i)

If $1\leq s\leq r\leq\infty$, then, for any $f\in L^{s}(\Omega)$,

$\left\|e^{td\Delta}f\right\|_{L^{r}}\leq
k_{1}\left(1+t^{-\frac{n}{2}\left(\frac{1}{s}-\frac{1}{r}\right)}\right)\left\|f\right\|_{L^{s}},\quad\forall\,t>0.$

  2. (ii)

If $1\leq s\leq r\leq\infty$, then, for all $f\in L^{s}(\Omega)$,

$\left\|\nabla e^{td\Delta}f\right\|_{L^{r}}\leq
k_{2}\left(1+t^{-\frac{1}{2}-\frac{n}{2}\left(\frac{1}{s}-\frac{1}{r}\right)}\right)e^{-\lambda_{1}t}\left\|f\right\|_{L^{s}},\quad\forall\,t>0.$

  3. (iii)

If $2\leq s\leq r<\infty$, then, for all $f\in W^{1,s}(\Omega)$,

$\left\|\nabla e^{td\Delta}f\right\|_{L^{r}}\leq
k_{3}\left(1+t^{-\frac{n}{2}\left(\frac{1}{s}-\frac{1}{r}\right)}\right)e^{-\lambda_{1}t}\left\|\nabla
f\right\|_{L^{s}},\quad\forall\,t>0.$

  4. (iv)

If $1<s\leq r\leq\infty$, then, for all $f\in\left(L^{s}(\Omega)\right)^{n}$,

$\left\|e^{td\Delta}\nabla\cdot f\right\|_{L^{r}}\leq
k_{4}\left(1+t^{-\frac{1}{2}-\frac{n}{2}\left(\frac{1}{s}-\frac{1}{r}\right)}\right)e^{-\lambda_{1}t}\left\|f\right\|_{L^{s}},\quad\forall\,t>0.$

Using variation-of-constants formula for $S$ and $I$, applying these semi-
group estimates as in Lemma (2.2), and then, invoking the Banach contraction
mapping principle and standard parabolic regularity of quasilinear parabolic
systems, one can readily derive the local-in-time solvability and
extensibility of the IBVP for the cross-diffusive SIS model (1.1); see similar
reasonings in [3, 45, 46, 55, 58].

######  Lemma 2.3.

Let $\Omega\subset\mathbb{R}^{n}\ (n\geq 1)$ be a bounded domain with a smooth
boundary, and suppose that the initial data $(S_{0},I_{0})$ satisfies (1.2).
Then there is a unique positive and classical maximal solution $(S,I)$ of the
IBVP (1.1) on some maximal interval $[0,T_{m})$ with $0<T_{m}\leq\infty$ such
that

$(S,I)\in\left(C(\overline{\Omega}\times[0,T_{m});W^{1,r_{0}}(\Omega))\cap
C^{2,1}(\overline{\Omega}\times(0,T_{m})\right)^{2}.$

Furthermore, if $T_{m}<\infty$, then

$\|S(\cdot,t)\|_{L^{\infty}}+\|I(\cdot,t)\|_{W^{1,r_{0}}}\to\infty\mbox{ as
}t\nearrow T_{m}.$ (2.2)

On $[0,T_{m})$, the $L^{1}$-norm of $S+I$ is non-increasing and hence is
uniformly bounded:

$\int_{\Omega}(S+I)\leq\int_{\Omega}(S_{0}+I_{0}).$ (2.3)

###### Proof.

As remarked above, the statements concerning the local-in-time existence of
classical solutions to the IBVP (1.1) and the extensibility criterion (2.2)
are well-established. The positivity of $(S,I)$ follows simply from the strong
maximum principle. Thanks to the homogeneous Neumann boundary conditions,
adding the $S$\- and $I$-equation in (1.1) and integrating by parts, we find
that

$\frac{d}{dt}\int_{\Omega}\left(S+I\right)=-\mu\int_{\Omega}I,$ (2.4)

which, upon integration, simply yields (2.3). ∎

For any $\tau\in(0,T_{m})$, we can shift the initial time $t=0$ to $t=\tau$ so
that upon retreating $(u(\cdot,\tau),v(\cdot,\tau))$ as the new initial data,
we will sometimes assume that $r_{0}=\infty$ in (1.2).

##  3\. A priori estimates and global well-posedness

In this section, unless otherwise specified, we will assume that the basic
conditions in the local existence of Lemma 2.3 hold. With such specifications,
we shall aim to derive $(L^{\infty},W^{1,r_{0}})$-type estimates of $(S,I)$
under two set of conditions: (I) $\chi$ is small and $p,q$ are relatively less
restrictive, and (II) $\chi$ is arbitrary and $p,q$ are relatively
restrictive. The small-$\chi$ boundedness is mainly based on Neumann semi-
group type estimates and the large-$\chi$ boundedness is based on mainly
coupled energy estimate method.

###  3.1. Small $\chi$-boundedness and global existence

In this subsection, we renovate a loop argument from [4] to provide
boundedness of classical solutions to (1.1) with $\chi$ small and $p,q$
relatively less restrictive.

######  Lemma 3.1.

Given $T\in(0,T_{m})$, and suppose there exists $M=M(T)>0$ such that

$\|S(\cdot,t)\|_{L^{\infty}}\leq M,\quad\quad\forall\,t\in(0,T).$ (3.1)

Let

$\displaystyle p<1+\min\left\\{1,\ \frac{2}{n}\right\\}.$

Then, for

$m=\begin{cases}\max\left\\{1,\,\frac{1}{p}\right\\},&\text{if }p<2,\
n=1,\\\\[5.69046pt] \frac{1}{p},&\text{if }p<\frac{2}{n},\ n\geq
2,\\\\[5.69046pt] \frac{n}{2-n\epsilon},&\text{if }\frac{2}{n}\leq
p<1+\frac{2}{n},\ n\geq 2,\end{cases}$ (3.2)

with $0<\epsilon<\min\left\\{\frac{2}{n},\ 1+\frac{2}{n}-p\right\\}$, one can
find $K_{1}>0$ independent of $M$ such that

$L:=\sup_{t\in(0,T)}\|I(\cdot,t)\|_{L^{\infty}}\leq
K_{1}\left(1+M^{\frac{qm}{m+1-pm}}\right)$ (3.3)

and, for any

$\zeta\in(0,1],\ \ s=\frac{nr_{0}}{n+(1-\zeta)r_{0}}\leq r_{0},$ (3.4)

there exists $K_{r_{0}}>0$ independent of $M,L$ such that

$\sup_{t\in(0,T)}\left\|\nabla I(\cdot,t)\right\|_{L^{r_{0}}}\leq
K_{r_{0}}\left[1+M^{q+\frac{qm}{m+1-pm}\left(p-\frac{1}{s}\right)^{+}}+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{s}\right)^{+}}\right].$
(3.5)

###### Proof.

In view of the variation-of-constants formula, we have

$0\leq I(\cdot,t)\leq e^{td_{I}(\Delta-
C_{0})}I_{0}+\int_{0}^{t}e^{(t-\tau)d_{I}(\Delta-C_{0})}(\beta
S^{q}I^{p})(\cdot,\tau)d\tau,$

where $C_{0}>0$ is such that $\gamma+\mu\geq C_{0}$ for all
$x\in\overline{\Omega}$ and $t>0$. Next, we apply (3.1), (3.2) and the
$L^{r}$-$L^{s}$-smoothing properties of the Neumann semigroup
$\\{e^{td_{I}\Delta}\\}_{t\geq 0}$ in Lemma 2.2 along with the maximum
principle to derive, for $t\in(0,T)$, that

$\begin{split}0\leq I(\cdot,t)&\leq\left\|e^{td_{I}(\Delta-
C_{0})}I_{0}\right\|_{L^{\infty}}+\int_{0}^{t}\left\|e^{(t-\tau)d_{I}(\Delta-
C_{0})}\left(\beta S^{q}I^{p}\right)(\cdot,\tau)\right\|_{L^{\infty}}d\tau\\\
&\leq\|I_{0}\|_{L^{\infty}}+C_{1}\int_{0}^{t}\left[1+(t-\tau)^{-\frac{n}{2m}}\right]e^{-C_{0}(t-\tau)}\left\|(S^{q}I^{p})(\cdot,\tau)\right\|_{L^{m}}d\tau.\end{split}$
(3.6)

We observe from (2.3), (3.2) and the definition of $N$ in (3.3) that, for
$\tau\in(0,T)$,

$\begin{split}&\|\beta(S^{q}I^{p})(\cdot,\tau)\|_{L^{m}}\leq\beta^{0}M^{q}\|I(\cdot,\tau)\|_{L^{pm}}^{p}\\\
&\leq\begin{cases}\beta^{0}M^{q}\|S_{0}+I_{0}\|_{L^{1}}^{\min\\{1,p\\}}N^{(p-1)^{+}},&\text{if
}n=1,\\\\[7.11317pt] \beta^{0}M^{q}\|S_{0}+I_{0}\|_{L^{1}},&\text{if
}p<\frac{2}{n},\ n\geq 2,\\\\[7.11317pt]
\beta^{0}M^{q}\|S_{0}+I_{0}\|_{L^{1}}^{\frac{1}{m}}N^{p-\frac{1}{m}},&\text{if
}\frac{2}{n}\leq p<1+\frac{2}{n},\ n\geq 2.\end{cases}\end{split}$ (3.7)

Thus, we infer from (3.6) and (3.7) that

$L\leq\begin{cases}C_{2}\left(1+M^{q}\right),&\text{if }\left\\{p\leq 1,\
n=1\right\\}\text{ or }\left\\{p<\frac{2}{n},\ n\geq
2\right\\},\\\\[7.11317pt]
C_{2}\left(1+M^{q}L^{p-\frac{1}{m}}\right),&\text{if }\left\\{1<p<2,\
n=1\right\\}\text{ or }\left\\{p\geq\frac{2}{n},\ n\geq
2\right\\}.\end{cases}$ (3.8)

In the first case of (3.8), observe that $pm=1$ and (3.3) holds automatically,
whereas in the second scenario, notice that the choice of $m$ in (3.2) implies
$p-\frac{1}{m}<1$, and then we use Young’s inequality with epsilon to deduce
from the second case of (3.8) that

$\displaystyle L$ $\displaystyle\leq C_{2}+C_{2}M^{q}L^{p-\frac{1}{m}}$
$\displaystyle\leq
C_{2}+\frac{1}{2}L+\frac{(m+1-pm)}{m}\left(\frac{2(pm-1)}{m}\right)^{\frac{pm-1}{m+1-pm}}\left(C_{2}M^{q}\right)^{\frac{m}{m+1-pm}},$

which quickly gives rise to our desired estimate (3.3).

Next, by the variation-of-constants formula, it holds

$\displaystyle
I(\cdot,t)=e^{td_{I}\Delta}I_{0}+\int_{0}^{t}e^{(t-\tau)d_{I}\Delta}(\beta
S^{q}I^{p})(\cdot,\tau)d\tau-\int_{0}^{t}e^{(t-\tau)d_{I}\Delta}((\gamma+\mu)I)(\cdot,\tau)d\tau.$

We take the gradient and then use the $L^{r}$-$L^{s}$-estimates of the Neumann
semigroup $\\{e^{td_{I}\Delta}\\}_{t\geq 0}$ in Lemma 2.2 and (3.3) to derive

$\begin{split}\left\|\nabla I(\cdot,t)\right\|_{L^{r_{0}}}&\leq\left\|\nabla
e^{td_{I}\Delta}I_{0}\right\|_{L^{r_{0}}}+\int_{0}^{t}\left\|\nabla
e^{(t-\tau)d_{I}\Delta}(\beta
S^{q}I^{p})(\cdot,\tau)\right\|_{L^{r_{0}}}d\tau\\\ &\ \
+\int_{0}^{t}\left\|\nabla
e^{(t-\tau)d_{I}\Delta}((\gamma+\mu)I)(\cdot,\tau)\right\|_{L^{r_{0}}}d\tau\\\
&\leq\|I_{0}\|_{W^{1,r_{0}}}+C_{3}\int_{0}^{t}\left[1+(t-\tau)^{-\frac{1}{2}-\frac{n}{2}\left(\frac{1}{s}-\frac{1}{r_{0}}\right)}\right]e^{-\lambda_{1}(t-\tau)}\left\|(S^{q}I^{p})(\cdot,t)\right\|_{L^{s}}d\tau\\\
&\ \
+C_{3}\int_{0}^{t}\left[1+(t-\tau)^{-\frac{1}{2}-\frac{n}{2}\left(\frac{1}{s}-\frac{1}{r_{0}}\right)}\right]e^{-\lambda_{1}(t-\tau)}\|I(\cdot,t)\|_{L^{s}}d\tau\\\
&\leq
C_{4}\left[1+M^{q}L^{\left(p-\frac{1}{s}\right)^{+}}+L^{\left(1-\frac{1}{s}\right)^{+}}\right]\int_{0}^{\infty}\left(1+\tau^{-1+\frac{\zeta}{2}}\right)e^{-\lambda_{1}\tau}d\tau,\end{split}$

which along with (3.3) immediately shows (3.5). ∎

With such preparations, we now close the loop in the following lemma.

######  Lemma 3.2.

Given $T\in(0,T_{m})$, and suppose that (3.1) and (3.2) hold. Then, for
$r>\frac{n}{2}$ and $r\geq 1$, one can find constant
$K_{3}>\|S_{0}\|_{L^{\infty}}$ independent of $M$ and $L$ such that

$\begin{split}\sup_{t\in(0,T)}\|S(\cdot,t)\|_{L^{\infty}}&\leq
K_{3}\Bigg{\\{}1+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r}\right)}\\\ &\ \ \ \ \
+M|\chi|\left[1+M^{q+\frac{qm}{m+1-pm}\left(p-\frac{1}{r_{0}}\right)^{+}}+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r_{0}}\right)}\right]\Bigg{\\}}.\end{split}$
(3.9)

###### Proof.

We first rewrite the $S$-equation via the semigroup representation as

$S(t)=e^{td_{S}\Delta}S_{0}+\chi\int_{0}^{t}e^{(t-\tau)d_{I}\Delta}\nabla\cdot(S\nabla
I)d\tau+\int_{0}^{t}e^{(t-\tau)d_{I}\Delta}\left(-\beta S^{q}I^{p}+\gamma
I\right)d\tau.$

Then we use (3.1), (3.3) and the well-known $L^{r}$-$L^{s}$-smoothing
properties of the Neumann semigroup $\\{e^{td_{I}\Delta}\\}_{t\geq 0}$ along
with the maximum principle to deduce, for $t\in(0,T)$, that

$\begin{split}S(\cdot,t)&\leq\left\|e^{td_{S}\Delta}S_{0}\right\|_{L^{\infty}}+|\chi|\int_{0}^{t}\left\|e^{(t-\tau)d_{S}\Delta}\nabla\cdot(S\nabla
I)(\cdot,\tau)\right\|_{L^{\infty}}d\tau\\\ &\ \ \
+\int_{0}^{t}\left\|e^{(t-\tau)d_{S}\Delta}\left(\gamma
I\right)(\cdot,\tau)\right\|_{L^{\infty}}d\tau\\\
&\leq\left\|S_{0}\right\|_{L^{\infty}}+C_{1}|\chi|\int_{0}^{t}\left[1+(t-\tau)^{-\frac{1}{2}-\frac{n}{2r_{0}}}\right]e^{-\lambda_{1}(t-\tau)}\left\|(S\nabla
I)(\cdot,\tau)\right\|_{L^{r_{0}}}d\tau\\\ &\ \ \
+C_{2}\int_{0}^{t}\left[1+(t-\tau)^{-\frac{n}{2r}}\right]e^{-\lambda_{1}(t-\tau)}\left\|I(\cdot,\tau)\right\|_{L^{r}}d\tau\\\
&\leq\|S_{0}\|_{L^{\infty}}+C_{3}|\chi|M\sup_{t\in(0,T)}\left\|\nabla
I(\cdot,t)\right\|_{L^{r_{0}}}+C_{4}N^{1-\frac{1}{r}},\end{split}$ (3.10)

which along with (3.3), (3.4) and (3.5) with $s=r_{0}$ yields the estimate
(3.9). ∎

Now, we are ready to establish our first result on the well-posedness of
system (1.1).

######  Theorem 3.3.

Let $\Omega\subset\mathbb{R}^{n}\ (n\geq 1)$ be a bounded and smooth domain.
Assume that

$np+(n-2)^{+}q<n+\min\left\\{n,\ 2\right\\}.$ (3.11)

Then, for any initial data $(S_{0},I_{0})$ fulfilling (1.2), there exists a
positive constant $\chi_{0}>0$ (cf. (3.14) below) such that whenever
$|\chi|<\chi_{0}$, the corresponding classical solution to the IBVP (1.1)
exists globally in time and is uniformly bounded in the sense there exists a
positive constant $C=C(S_{0},I_{0},\chi,\Omega,p,q,\beta,\gamma,\mu)>0$ such
that

$\|S(\cdot,t)\|_{L^{\infty}}+\|I(\cdot,t)\|_{W^{1,r_{0}}}\leq C,\ \ \
\forall\,t>0.$ (3.12)

###### Proof.

Based on (3.2) and (3.11), we specify $m$ and $r$ as follows:

$\begin{cases}m=\max\left\\{1,\ \frac{1}{p}\right\\},\ \ r=1,&\text{if
}n=1,\\\\[5.69046pt] m=\frac{1}{p},\ \
r=\frac{1}{2}\left[\frac{n}{2}+\min\left\\{\frac{q}{(q-1)^{+}},\
n\right\\}\right],&\text{if }n\geq 2,\ p<\frac{2}{n},\\\\[5.69046pt]
m=\frac{n}{2-n\epsilon},\ \
r=\frac{1}{2}\left[\frac{n}{2}+\min\left\\{\frac{q}{\left[q-\left(1+\frac{2}{n}-p\right)+\epsilon\right]^{+}},\
n\right\\}\right],&\text{if }n\geq 2,\ \frac{2}{n}\leq
p<1+\frac{2}{n},\end{cases}$

where the positive constant $\epsilon$ is given by

$\epsilon=\frac{1}{2}\min\left\\{\frac{2}{n},\
1+\frac{2}{n}-p-\left(1-\frac{2}{n}\right)q\right\\}>0,\ \ n\geq 2.$

Then by plain verifications we find that both $m$ and $r$ satisfy the
conditions set forth in Lemmas 3.1 and 3.2, and moreover that

$\frac{qm}{m+1-pm}\left(1-\frac{1}{r}\right)<1.$ (3.13)

For those geometrical constants $K_{i}$ depending on the initial data and
$\Omega$ provided by Lemmas 3.1 and 3.2, thanks to (3.13), we first define a
positive and finite constant $\chi_{0}$ by

$\begin{split}\chi_{0}&=\sup_{z>0}\frac{z-K_{3}\left[1+z^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r}\right)}\right]}{K_{3}z\left[1+z^{q+\frac{qm}{m+1-pm}\left(p-\frac{1}{r_{0}}\right)^{+}}+z^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r_{0}}\right)}\right]}\\\
&\ \
=\frac{M-K_{3}\left[1+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r}\right)}\right]}{K_{3}M\left[1+M^{q+\frac{qm}{m+1-pm}\left(p-\frac{1}{r_{0}}\right)^{+}}+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r_{0}}\right)}\right]}\in(0,\infty)\end{split}$
(3.14)

for some $M>K_{3}$. This, along with the fact
$K_{3}\geq\|S_{0}\|_{L^{\infty}}$, necessarily implies that

$\|S_{0}\|_{L^{\infty}}<M,\quad\mbox{and}\quad
K_{3}\left(1+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r}\right)}\right)<M.$ (3.15)

Now, for $\chi\in[-\chi_{0},\chi_{0}]$, we let $(S,I)$ be the corresponding
maximally extended solution of the IBVP (1.1) on $\Omega\times(0,T_{m})$ with
$T_{m}\in(0,\infty]$. Thanks to Lemmas 3.1 and 3.2, we next employ a standard
continuity argument to close the loop-argument and then conclude global
existence (i.e., $T_{m}=\infty$) and boundedness (3.12). To this end, we
define

$\mathcal{U}:=\Bigl{\\{}(0,T_{0})\subset(0,T_{m}):\
\|S(\cdot,t)\|_{L^{\infty}}\leq M,\ \forall\,t\in(0,T_{0})\Bigr{\\}}.$ (3.16)

Then the continuity of $S$ and (3.15) imply that $\mathcal{U}$ is nonempty and
so $T=\sup\mathcal{U}$ is well-defined and $T\in(0,\infty]$. Furthermore, by
(3.9), (3.14) and (3.15), we infer that

$\begin{split}\sup_{t\in(0,T)}\|S(\cdot,t)\|_{L^{\infty}}&\leq
K_{3}\left[1+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r}\right)}\right]\\\ &\ \ \
\ \
+K_{3}M|\chi|\left[1+M^{q+\frac{qm}{m+1-pm}\left(p-\frac{1}{r_{0}}\right)^{+}}+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r_{0}}\right)}\right]\\\
&\leq M.\end{split}$ (3.17)

Finally, combining (3.17), (3.3) and (3.5) and the extensibility criterion in
(2.2) of Lemma 2.3, we must have that $T_{m}=\infty$ and and then we achieve
our claimed uniform $(L^{\infty},W^{1,r_{0}})$-boundedness of $(S,I)$ on
$\Omega\times(0,\infty)$. ∎

###  3.2. Large $\chi$-boundedness and global existence

In this subsection, we shall present two types of large $\chi$-boundedness of
solutions to (1.1) via mainly semi-group method and energy estimate method as
illustrated in Subsections 3.2.1 and 3.2.2.

####  3.2.1. Large $\chi$-boundedness in any dimensions with $p,q$ restrictive

We first use arguments in sprit similar to Subsection 3.1 to obtain the
following large $\chi$-boundedness in any dimensions while with more
restrictive $p$ and $q$.

######  Theorem 3.4.

Let $\Omega\subset\mathbb{R}^{n}\ (n\geq 1)$ be a bounded and smooth domain.
Assume that

$q<\frac{1}{n+1}\quad\mbox{and}\quad\
p+(n+1)q<1+\min\left\\{1,\frac{2}{n}\right\\}.$ (3.18)

Then, for any initial data $(S_{0},I_{0})$ fulfilling (1.2), the corresponding
classical solution to the IBVP (1.1) exists globally in time and is uniformly
bounded in the sense of (3.12).

###### Proof.

Taking $r_{0}=\infty$ and $s=\frac{n}{1-\zeta}$ in (3.4) with $\zeta\in(0,1]$,
we get from (3.5) that

$\sup_{t\in(0,T)}\|\nabla I(\cdot,t)\|_{L^{\infty}}\leq
K_{\zeta}\left[1+M^{q+\frac{qm}{m+1-pm}\left(p-\frac{1}{n}+\frac{\zeta}{n}\right)^{+}}+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{n}+\frac{\zeta}{n}\right)^{+}}\right].$
(3.19)

With this, for any $r>\frac{n}{2}$ and $r\geq 1$, we re-bound (3.10) as
follows:

$\begin{split}S(\cdot,t)&\leq\left\|e^{td_{S}\Delta}S_{0}\right\|_{L^{\infty}}+|\chi|\int_{0}^{t}\left\|e^{(t-\tau)d_{S}\Delta}\nabla\cdot(S\nabla
I)(\cdot,\tau)\right\|_{L^{\infty}}d\tau\\\ &\ \ \
+\int_{0}^{t}\left\|e^{(t-\tau)d_{S}\Delta}\left(\gamma
I\right)(\cdot,\tau)\right\|_{L^{\infty}}d\tau\\\
&\leq\|S_{0}\|_{L^{\infty}}+C_{1}|\chi|\int_{0}^{t}\left[1+(t-\tau)^{-\frac{1}{2}-\frac{n}{4r}}\right]e^{-\lambda_{1}(t-\tau)}\left\|(S\nabla
I)(\cdot,\tau)\right\|_{L^{2r}}d\tau\\\ &\ \ \
+C_{2}\int_{0}^{t}\left[1+(t-\tau)^{-\frac{n}{2r}}\right]e^{-\lambda_{1}(t-\tau)}\|I(\cdot,\tau)\|_{L^{r}}d\tau\\\
&\leq\|S_{0}\|_{L^{\infty}}+C_{3}|\chi|M^{1-\frac{1}{2r}}\sup_{t\in(0,T)}\|\nabla
I(\cdot,t)\|_{L^{\infty}}+C_{4}N^{1-\frac{1}{r}},\end{split}$ (3.20)

which along with (3.19) and (3.3) allows us to infer

$\begin{split}\sup_{t\in(0,T)}\|S(\cdot,t)\|_{L^{\infty}}&\leq
C_{5}\Bigg{[}1+|\chi|M^{1-\frac{1}{2r}}+|\chi|M^{q+1-\frac{1}{2r}+\frac{qm}{m+1-pm}\left(p-\frac{1}{n}+\frac{\zeta}{n}\right)^{+}}\\\
&\ \ \
+|\chi|M^{1-\frac{1}{2r}+\frac{qm}{m+1-pm}\left(1-\frac{1}{n}+\frac{\zeta}{n}\right)^{+}}+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r}\right)}\Bigg{]}.\end{split}$
(3.21)

Now, we observe that (3.18) implies that

$\left\\{\begin{array}[]{lll}q+1-\frac{1}{n}+\frac{q\frac{n}{2}}{\frac{n}{2}+1-p\frac{n}{2}}\left(p-\frac{1}{n}\right)^{+}<1,\\\\[8.5359pt]
1-\frac{1}{n}+\frac{q\frac{n}{2}}{\frac{n}{2}+1-p\frac{n}{2}}\left(1-\frac{1}{n}\right)^{+}<1,\\\\[8.5359pt]
\frac{q\frac{n}{2}}{\frac{n}{2}+1-p\frac{n}{2}}\left(1-\frac{2}{n}\right)^{+}<1,\end{array}\right.$

which upon continuity allows us to fix $0<\epsilon<\min\left\\{\frac{2}{n},\
1+\frac{2}{n}-p\right\\}$ (cf. (3.2)) such that

$\begin{cases}q+1-\frac{1}{n+\epsilon}+\frac{q\frac{n}{2-n\epsilon}}{\frac{n}{2-n\epsilon}+1-p\frac{n}{2-n\epsilon}}\left(p-\frac{1-\epsilon}{n}\right)^{+}<1,\\\\[7.11317pt]
1-\frac{1}{n+\epsilon}+\frac{q\frac{n}{2-n\epsilon}}{\frac{n}{2-n\epsilon}+1-p\frac{n}{2-n\epsilon}}\left(1-\frac{1-\epsilon}{n}\right)^{+}<1,\\\\[7.11317pt]
\frac{q\frac{n}{2-n\epsilon}}{\frac{n}{2-n\epsilon}+1-p\frac{n}{2-n\epsilon}}\left(1-\frac{2}{n+\epsilon}\right)^{+}<1.\end{cases}$
(3.22)

Based on (3.18) and (3.21), we first choose $m$, $r$ and $\zeta$ as follows:

$\begin{cases}m=\frac{1}{p},\ \ r=1,\ \ \zeta=\frac{1}{2}\min\left\\{1-p,\
\frac{1}{2q}\right\\},&\text{if }n=1,\ p<1,\\\\[5.69046pt] m=1,\ \ r=1,\ \
\zeta=\frac{2-p-2q}{3q},&\text{if }n=1,\ p\geq 1,\\\\[5.69046pt]
m=\frac{1}{p},\
r=\frac{1}{2}\left\\{\frac{n}{2}+\frac{n}{2q[(np-1)^{+}+n]}\right\\},&\text{if
}n\geq 2,\ p<\frac{2}{n},\\\\[5.69046pt] m=\frac{n}{2-n\zeta},\
r=\frac{n+\zeta}{2},\ \zeta=\epsilon,&\text{if }n\geq 2,\
p\geq\frac{2}{n}.\end{cases}$ (3.23)

In the third case of (3.23), $\zeta>0$ is chosen so that

$\left\\{\begin{array}[]{ll}p-\frac{1}{n}+\frac{\zeta}{n}<0,&\text{if
}p<\frac{1}{n},\\\\[7.11317pt]
\left(np-1+n+\zeta\right)\left[\frac{q}{2}+\frac{1}{2(np-1+n)}\right]<1,&\text{if
}p\geq\frac{1}{n},\end{array}\right.$

which is feasible since

$\displaystyle\left(np-1+n\right)\left[\frac{q}{2}+\frac{1}{2(np-1+n)}\right]=\frac{1}{2}+\frac{q}{2}(np-1+n)<\frac{1}{2}+\frac{q}{2}(n+1)<1,$

for $p\geq\frac{1}{n}$ and $(p,q)$ fulfilling (3.18). With such chosen $m$,
$r$ and $\zeta$, we then compute from (3.21), (3.22) and (3.23) that

$\begin{cases}q+1-\frac{1}{2r}+\frac{qm}{m+1-pm}\left(p-\frac{1}{n}+\frac{\zeta}{n}\right)^{+}<1,\\\\[5.69046pt]
1-\frac{1}{2r}+\frac{qm}{m+1-pm}\left(1-\frac{1}{n}+\frac{\zeta}{n}\right)^{+}<1,\\\\[5.69046pt]
\frac{qm}{m+1-pm}\left(1-\frac{1}{r}\right)<1.\end{cases}$ (3.24)

In light of (3.24), one can easily use Young’s inequality with epsilon to
(3.21) to derive the existence of
$M=M(S_{0},I_{0},\beta,\gamma,\mu,\Omega,\chi,,p,q)>\|S_{0}\|_{L^{\infty}}$
such that

$\begin{split}\sup_{t\in(0,T)}\|S(\cdot,t)\|_{L^{\infty}}&\leq
C_{5}\Bigr{[}1+|\chi|M^{1-\frac{1}{2r}}+|\chi|M^{q+1-\frac{1}{2r}+\frac{qm}{m+1-pm}\left(p-\frac{1}{n}+\frac{\zeta}{n}\right)^{+}}\\\
&\ \
\quad+|\chi|M^{1-\frac{1}{2r}+\frac{qm}{m+1-pm}\left(1-\frac{1}{n}+\frac{\zeta}{n}\right)^{+}}+M^{\frac{qm}{m+1-pm}\left(1-\frac{1}{r}\right)}\Bigr{]}\\\
&\leq M.\end{split}$

With this, we see the set defined by (3.16) is a nonempty and both open and
closed set, and thus, using similar continuity arguments as done in Theorem
3.3, we readily obtain that $T_{m}=\infty$ and that $(S,I)$ is uniform-in-time
bounded according to (3.12). ∎

####  3.2.2. Large $\chi$-boundedness in lower dimentions

In this subsection, we shall employ coupled subtle energy estimates to improve
the boundedness range in (3.18) for $n\leq 2$.

To move from $L^{1}$-boundness obtained in (2.3) to higher order regularity
for the local solution $(S,I)$ of (1.1), we begin to study the time evolutions
of $\|(S+1)\ln(S+1)\|_{L^{1}}$ and $\|\nabla I\|_{L^{2}}^{2}$; upon
integration by parts on (1.1), we find the following energy identities.

######  Lemma 3.5.

The local-in-time solution of (1.1) satisfies

$\begin{split}&\frac{d}{dt}\int_{\Omega}(S+1)\ln(S+1)+4d_{S}\int_{\Omega}\left|\nabla(S+1)^{\frac{1}{2}}\right|^{2}+\beta\int_{\Omega}S^{q}I^{p}[\ln(S+1)+1]\\\
&\ \ =\chi\int_{\Omega}[S-\ln(S+1)]\Delta
I+\gamma\int_{\Omega}I[\ln(S+1)+1]\end{split}$ (3.25)

and

$\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
I|^{2}+(\gamma+\mu)\int_{\Omega}|\nabla I|^{2}+d_{I}\int_{\Omega}|\Delta
I|^{2}=-\beta\int_{\Omega}S^{q}I^{p}\Delta I.$ (3.26)

To bound the bad terms on the right-hand sides of (3.25) and (3.26) in terms
of the dissipation terms on their left-hand sides, based on the
$L^{1}$-boundedness of $S+I$ in (2.3), we employ the G-N inequality (cf. Lemma
2.1) to deduce the following estimates.

######  Lemma 3.6.

(i) Let $\Omega\subset\mathbb{R}^{1}$ be a finite interval. Then, for
$r\in(1,\infty]$ and $\epsilon>0$, it holds

$\begin{cases}\left\|S+1\right\|_{L^{r}}^{\frac{2r}{r-1}}\leq
C\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C,\\\\[7.11317pt]
\forall s<\frac{2r}{r-1},\ \
\left\|S+1\right\|_{L^{r}}^{s}\leq\epsilon\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C_{\epsilon},\\\\[7.11317pt]
\left\|(S+1)\ln(S+1)\right\|_{L^{1}}\leq\epsilon\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C_{\epsilon}\end{cases}$
(3.27)

and

$\begin{cases}\|I\|_{L^{r}}^{\frac{5r}{r-1}}\leq C\|\Delta
I\|_{L^{2}}^{2}+C\|\nabla I\|_{L^{2}}^{2}+C,\\\\[7.11317pt] \forall
s<\frac{5r}{r-1},\ \ \|I\|_{L^{r}}^{s}\leq\epsilon\|\Delta
I\|_{L^{2}}^{2}+\epsilon\|\nabla I\|_{L^{2}}^{2}+C_{\epsilon}.\end{cases}$
(3.28)

(ii) Let $\Omega\subset\mathbb{R}^{2}$ be bounded and smooth. Then, for
$r\in(1,\infty)$ and $\epsilon>0$, it holds

$\begin{cases}\|S+1\|_{L^{r}}^{\frac{r}{r-1}}\leq
C\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C,\\\\[7.11317pt]
\forall s<\frac{r}{r-1},\ \
\|S+1\|_{L^{r}}^{s}\leq\epsilon\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C_{\epsilon},\\\\[7.11317pt]
\|(S+1)\ln(S+1)\|_{L^{1}}\leq\epsilon\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C_{\epsilon}\end{cases}$
(3.29)

and, for $r\in(1,\infty]$ and $\epsilon>0$, it holds

$\begin{cases}\|I\|_{L^{r}}^{\frac{3r}{r-1}}\leq C\|\Delta
I\|_{L^{2}}^{2}+C\|\nabla I\|_{L^{2}}^{2}+C,\\\\[7.11317pt] \forall
s<\frac{3r}{r-1},\ \ \|I\|_{L^{r}}^{s}\leq\epsilon\|\Delta
I\|_{L^{2}}^{2}+\epsilon\|\nabla I\|_{L^{2}}^{2}+C_{\epsilon}.\end{cases}$
(3.30)

###### Proof.

(i) By the $L^{1}$-bound of $S$ in (2.3), we apply the 1-D G-N inequality as
stated in Lemma 2.1 to infer, for $r\in(1,\infty]$,

$\displaystyle\|S+1\|_{L^{r}}^{s}$
$\displaystyle=\left\|(S+1)^{\frac{1}{2}}\right\|_{L^{2r}}^{2s}$
$\displaystyle\leq
C\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{\frac{(r-1)s}{r}}\left\|(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{\frac{(r+1)s}{r}}+C\left\|(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2s}$
$\displaystyle\leq
C\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{\frac{(r-1)s}{r}}+C$
$\displaystyle\begin{cases}=C\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C,&\text{
if }s=\frac{2r}{r-1},\\\\[7.11317pt]
\leq\epsilon\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C_{\epsilon},&\text{
if }s<\frac{2r}{r-1}.\end{cases}$

This, together with the simple algebraic fact that

$\|(S+1)\ln(S+1)\|_{L^{1}}\leq\|S+1\|_{L^{\frac{3}{2}}}^{\frac{3}{2}},$ (3.31)

shows the desired estimates stated in (3.27).

As for the bounds for $I$ in (3.28), in view of the $L^{1}$-boundedness of $I$
guaranteed by (2.3), we use the 1-D G-N inequality to derive, for
$r\in(1,\infty]$, that

$\begin{split}\|I\|_{L^{r}}^{s}&\leq
C\left\|D^{2}I\right\|_{L^{2}}^{\frac{2(r-1)s}{5r}}\left\|I\right\|_{L^{1}}^{\frac{(3r+2)s}{5r}}+C\|I\|_{L^{1}}^{s}\\\
&\leq C\left\|D^{2}I\right\|_{L^{2}}^{\frac{2(r-1)s}{5r}}+C\\\
&\begin{cases}=C\left\|D^{2}I\right\|_{L^{2}}^{2}+C,&\text{ if
}s=\frac{5r}{r-1},\\\\[7.11317pt]
\leq\epsilon\left\|D^{2}I\right\|_{L^{2}}^{2}+C_{\epsilon},&\text{ if
}s<\frac{5r}{r-1}.\end{cases}\end{split}$ (3.32)

In a similar manner, we also have, for bounded smooth
$\Omega\subset\mathbb{R}^{n}(n\geq 1)$ and any $\epsilon>0$,

$\|I\|_{L^{2}}^{2}\leq\epsilon\|\nabla I\|_{L^{2}}^{2}+C_{\epsilon}.$ (3.33)

On the other hand, for bounded smooth $\Omega\subset\mathbb{R}^{n}(n\geq 1)$,
the $H^{2}$-regularity shows

$\|D^{2}I\|_{L^{2}}^{2}\leq C\|\Delta I\|_{L^{2}}^{2}+C\|I\|_{L^{2}}^{2}.$
(3.34)

Joining this with (3.32) and (3.33), we end up with the estimates claimed in
(3.28).

(ii) By the $L^{1}$-bound of $S$ in (2.3), we apply the 2D G-N inequality as
stated in Lemma 2.1 to obatin, for $r\in(1,\infty)$,

$\displaystyle\|S+1\|_{L^{r}}^{s}$
$\displaystyle=\left\|(S+1)^{\frac{1}{2}}\right\|_{L^{2r}}^{2s}$
$\displaystyle\leq
C\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{\frac{2(r-1)s}{r}}\left\|(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{\frac{2s}{r}}+C\left\|(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2s}$
$\displaystyle\leq
C\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{\frac{2(r-1)s}{r}}+C$
$\displaystyle\begin{cases}=C\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C,&\text{
if }s=\frac{r}{r-1},\\\\[7.11317pt]
\leq\epsilon\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C_{\epsilon},&\text{
if }s<\frac{r}{r-1}.\end{cases}$

This together with the simple fact (3.31) entails the desired estimates stated
in (3.29).

To achieve the bounds for $I$ in (3.30), we use the $L^{1}$-boundedness of $I$
ensured by (2.3) and the 2D G-N inequality to deduce, for $r\in(1,\infty]$,
that

$\displaystyle\|I\|_{L^{r}}^{s}$ $\displaystyle\leq
C\left\|D^{2}I\right\|_{L^{2}}^{\frac{2(r-1)s}{3r}}\|I\|_{L^{1}}^{\frac{(r+2)s}{3r}}+C\|I\|_{L^{1}}^{s}$
$\displaystyle\leq C\left\|D^{2}I\right\|_{L^{2}}^{\frac{2(r-1)s}{3r}}+C$
$\displaystyle\begin{cases}=C\left\|D^{2}I\right\|_{L^{2}}^{2}+C,&\text{ if
}s=\frac{3r}{r-1},\\\\[7.11317pt]
\leq\epsilon\left\|D^{2}I\right\|_{L^{2}}^{2}+C_{\epsilon},&\text{ if
}s<\frac{3r}{r-1}.\end{cases}$

This along with (3.33) and (3.34) yields easily (3.30). ∎

Thanks to the $L^{1}$-boundedness of $S+I$ in (2.3) and Lemma 3.6, we now
deduce higher order regularity for $(S,I)$ under certain restrictions of $q$
and $p$.

######  Lemma 3.7.

Let $q$ and $p$ satisfy

$\begin{cases}10q+4p<15,\ \ \ \ q+p<3,&\text{ if }n=1,\\\\[7.11317pt] 3q+p<3,\
\ \ \ q+p<2,&\text{ if }n=2.\end{cases}$ (3.35)

Then the local-in-time solution of (1.1) satisfies

$\|(S+1)\ln(S+1)\|_{L^{1}}+\|\nabla I\|_{L^{2}}\leq
C,\quad\quad\forall\,t\in(0,T_{m}).$ (3.36)

Moreover, there exists $C>0$ such that

$\|I(\cdot,t)\|_{L^{\infty}}\leq C,\quad\quad\forall\,t\in(0,T_{m}).$ (3.37)

###### Proof.

Due to the $L^{1}$-boundedness of $S+I$, Lemma 3.6 and (3.33), we bound the
bad terms on the right-hand sides of (3.25) and (3.26) in terms of the
dissipation terms on their left-hand sides in the following manners:

$\begin{split}\chi\int_{\Omega}[S-\ln(S+1)]\Delta
I&\leq\epsilon\int_{\Omega}[S-\ln(S+1)]^{2}+\frac{\chi^{2}}{4\epsilon}\int_{\Omega}|\Delta
I|^{2}\\\
&\leq\epsilon\int_{\Omega}(S+1)^{2}+\frac{\chi^{2}}{4\epsilon}\int_{\Omega}|\Delta
I|^{2}\\\ &\leq
C\epsilon\int_{\Omega}\left|\nabla(S+1)^{\frac{1}{2}}\right|^{2}+C\epsilon+\frac{\chi^{2}}{4\epsilon}\int_{\Omega}|\Delta
I|^{2},\end{split}$ (3.38)
$\begin{split}\gamma\int_{\Omega}I[\ln(S+1)+1]&\leq\gamma\int_{\Omega}\ln^{2}(S+1)+\gamma\int_{\Omega}I^{2}+\gamma\int_{\Omega}I\\\
&\leq\gamma\int_{\Omega}(S+I)+\gamma\int_{\Omega}I^{2}\\\
&\leq\gamma\int_{\Omega}(S_{0}+I_{0})+\gamma\epsilon\int_{\Omega}|\nabla
I|^{2}+\gamma C_{\epsilon}\end{split}$ (3.39)

and

$-\beta\int_{\Omega}S^{q}I^{p}\Delta I\leq\frac{d_{I}}{4}\int_{\Omega}|\Delta
I|^{2}+\frac{\beta^{2}}{d_{I}}\int_{\Omega}S^{2q}I^{2p}.$ (3.40)

In the sequel, we apply Hölder inequality and Young’s inequality with epsilon
to estimate the second integral on the right-hand of (3.40) as follows: In the
simple case of

$q+p\leq\frac{1}{2},$ (3.41)

we have from Hölder inequality that

$\int_{\Omega}S^{2q}I^{2p}\leq\|S\|_{L^{1}}^{2q}\|I\|_{L^{\frac{2p}{1-2q}}}^{2p}\leq\|S\|_{L^{1}}^{2q}\|I\|_{L^{1}}^{2p}|\Omega|^{(1-2q-2p)}\leq\|S_{0}+I_{0}\|_{L^{1}}|\Omega|^{(1-2q-2p)}.$
(3.42)

In the opposite scenario of (3.41), we break our argument into two cases:

Case of $n=1$. Because of (3.35), we can take $r\in(1,\infty)$ such that

$\max\left\\{1,\ \ 2p,\ \
\frac{6p}{15-10q-4p}\right\\}<r<\min\left\\{\frac{2p}{(2p-5)^{+}},\ \
\frac{2p}{(1-2q)^{+}}\right\\},$

which imply

$r>1,\ r>2p,\ 2p<\frac{5r}{r-1},\ \frac{2qr}{r-2p}=:s>1,\
2q<\frac{10qr}{5r-2p(r-1)}<\frac{2s}{s-1};$

with such chosen $r$, we utilize Hölder and Young’s inequality with epsilon,
(3.27) and (3.28) to estimate, for any $\epsilon,\eta>0$, that

$\begin{split}\int_{\Omega}S^{2q}I^{2p}&\leq\left\|S+1\right\|_{L^{\frac{2qr}{r-2p}}}^{2q}\|I\|_{L^{r}}^{2p}\\\
&\leq\epsilon\|I\|_{L^{r}}^{\frac{5r}{r-1}}+C_{\epsilon}\left\|S+1\right\|_{L^{\frac{2qr}{r-2p}}}^{\frac{10qr}{5r-2p(r-1)}}\\\
&\leq C\epsilon\|\Delta I\|_{L^{2}}^{2}+C\epsilon\|\nabla
I\|_{L^{2}}^{2}+C\epsilon+\eta\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C_{\epsilon,\eta}.\end{split}$
(3.43)

Case of $n=2$. In this case, under (3.35), we will show that (3.43) type
estimate also holds. Indeed, because of (3.35), we can take $r\in(1,\infty)$
such that

$\max\left\\{1,\ \ 2p,\ \
\frac{2p}{3-3q-p}\right\\}<r<\min\left\\{\frac{2p}{(2p-3)^{+}},\ \
\frac{2p}{(1-2q)^{+}}\right\\},$

which imply

$r>1,\ \ r>2p,\ \ 2p<\frac{3r}{r-1},\ \ \frac{2qr}{r-2p}=:s>1,\ \
2q<\frac{6qr}{3r-2p(r-1)}<\frac{s}{s-1};$

with such chosen $r$, we utilize Hölder and Young’s inequality with epsilon,
(3.29) and (3.30) to infer, for any $\epsilon,\eta>0$, that

$\begin{split}\int_{\Omega}S^{2q}I^{2p}&\leq\|S+1\|_{L^{\frac{2qr}{r-2p}}}^{2q}\|I\|_{L^{r}}^{2p}\\\
&\leq\epsilon\|I\|_{L^{r}}^{\frac{3r}{r-1}}+C_{\epsilon}\|S+1\|_{L^{\frac{2qr}{r-2p}}}^{\frac{6qr}{3r-2p(r-1)}}\\\
&\leq C\epsilon\|\Delta I\|_{L^{2}}^{2}+C\epsilon\|\nabla
I\|_{L^{2}}^{2}+C\epsilon+\eta\left\|\nabla(S+1)^{\frac{1}{2}}\right\|_{L^{2}}^{2}+C_{\epsilon,\eta}.\end{split}$
(3.44)

In summary, we substitute (3.38), (3.39), (3.40), (3.43), (3.42) and (3.44)
into (3.25) and (3.26) to conclude, for any $\epsilon,\eta>0$, there exist
constants $M,M_{\epsilon},L,L_{\eta}>0$ such that

$\frac{d}{dt}\int_{\Omega}(S+1)\ln(S+1)+3d_{S}\int_{\Omega}\left|\nabla(S+1)^{\frac{1}{2}}\right|^{2}\leq\gamma\epsilon\int_{\Omega}|\nabla
I|^{2}+M\chi^{2}\int_{\Omega}|\Delta I|^{2}+M_{\epsilon},$ (3.45)

and

$\frac{d}{dt}\int_{\Omega}|\nabla I|^{2}+(\gamma+\mu)\int_{\Omega}|\nabla
I|^{2}+d_{I}\int_{\Omega}|\Delta I|^{2}\leq
L\eta\int_{\Omega}\left|\nabla(S+1)^{\frac{1}{2}}\right|^{2}+L_{\eta}.$ (3.46)

Multiplying (3.45) by $d_{I}$ and (3.46) by $2M\chi^{2}$ and then choosing
sufficiently small $\epsilon$ and $\eta$ and then using Lemma 3.6, we finally
end up with, for some $K>0$,

$\begin{split}&\frac{d}{dt}\int_{\Omega}\left[d_{I}(S+1)\ln(S+1)+2M\chi^{2}|\nabla
I|^{2}\right]+d_{I}\int_{\Omega}(S+1)\ln(S+1)\\\ &\ \
+(\gamma+\mu)M\chi^{2}\int_{\Omega}|\nabla
I|^{2}+d_{S}d_{I}\int_{\Omega}\left|\nabla(S+1)^{\frac{1}{2}}\right|^{2}+d_{I}M\chi^{2}\int_{\Omega}|\Delta
I|^{2}\leq K.\end{split}$

Solving this standard Grownall inequality, we directly obtain the uniform
estimate (3.36).

In the case of $n=1$, by (3.36) and (3.33), we know that the $W^{1,2}$-norm of
$I$ is uniformly bounded, and thus (3.37) follows from the continuous
embedding of $W^{1,2}(\Omega)$ into $L^{\infty}(\Omega)$.

Next, in the case of $n=2$, the $W^{1,2}$-boundedness of $I$ implied by (3.36)
and (3.33) yield first the uniform boundedness of $\|I\|_{L^{q}}$ for any
$q\in(0,\infty)$:

$\|I(\cdot,t)\|_{L^{q}}\leq C_{q},\ \ \ \forall\,t\in(0,T_{m}).$ (3.47)

To show the $L^{\infty}$-boundedness of $I$, we use semi-group representation
to rewrite $I$ as

$I=e^{t(d_{I}\Delta-(\gamma+\mu))}I_{0}+\beta\int_{0}^{t}e^{(t-s)(d_{I}\Delta-(\gamma+\mu))}S^{q}I^{p}ds.$
(3.48)

Based on the $L^{r}$-$L^{s}$ type estimates in Lemma 2.2, the
$L^{1}$-boundedness of $S$, (3.47) and the fact that $q<1$ ensured by (3.35),
we apply Hölder interpolation inequality to estimate

$\displaystyle\|I(\cdot,t)\|_{L^{\infty}}$
$\displaystyle\leq\left\|e^{t(d_{I}\Delta-(\gamma+\mu))}I_{0}\right\|_{L^{\infty}}+\beta\int_{0}^{t}\left\|e^{(t-s)(d_{I}\Delta-(\gamma+\mu))}S^{q}I^{p}\right\|_{L^{\infty}}ds$
$\displaystyle\leq
C\|I_{0}\|_{L^{\infty}}+C\int_{0}^{t}\left[1+(t-s)^{-\frac{2q}{q+1}}\right]e^{-(\gamma+\mu)(t-s)}\left\|S^{q}I^{p}\right\|_{L^{\frac{q+1}{2q}}}ds$
$\displaystyle\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{2q}{q+1}}\right]e^{-(\gamma+\mu)(t-s)}\|S^{q}\|_{L^{\frac{1}{q}}}\|I^{p}\|_{L^{\frac{q+1}{(1-q)q}}}ds$
$\displaystyle\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{2q}{q+1}}\right]e^{-(\gamma+\mu)(t-s)}ds$
$\displaystyle\leq
C+C\int_{0}^{\infty}\left(1+\tau^{-\frac{2q}{q+1}}\right)e^{-(\gamma+\mu)\tau}d\tau\leq
C,$

which precisely gives rise to (3.37). ∎

Based on the basic but crucial starting boundedness provided in Lemma 3.7, we
use test-procedure to obtain the $L^{2}$-bound for $S$ and $|\nabla I|^{4}$ by
estimating the combined time evolution of $\|S\|_{L^{2}}^{2}$ and $\|\nabla
I\|_{L^{4}}^{4}$.

######  Lemma 3.8.

Let (3.35) hold. Then the local-in-time solution of (1.1) satisfies

$\|S(\cdot,t)\|_{L^{2}}+\|\nabla I(\cdot,t)\|_{L^{4}}\leq
C,\quad\quad\forall\,t\in(0,T_{m}).$ (3.49)

###### Proof.

We test the $S$-equation in (1.1) by $S$ and then integrate by parts to derive
that

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}S^{2}+d_{S}\int_{\Omega}|\nabla
S|^{2}+\beta\int_{\Omega}S^{1+q}I^{p}$
$\displaystyle=-\chi\int_{\Omega}S\nabla S\cdot\nabla I+\gamma\int_{\Omega}SI$
$\displaystyle\leq\frac{d_{S}}{4}\int_{\Omega}|\nabla
S|^{2}+\frac{\chi^{2}}{d_{S}}\int_{\Omega}S^{2}|\nabla
I|^{2}+\gamma\|I\|_{L^{\infty}}\int_{\Omega}S,$

which, together with (2.3) and (3.37), gives

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}S^{2}+\frac{3d_{S}}{4}\int_{\Omega}|\nabla
S|^{2}+\beta\int_{\Omega}S^{1+q}I^{p}\leq\frac{\chi^{2}}{d_{S}}\int_{\Omega}S^{2}|\nabla
I|^{2}+C_{1}.$ (3.50)

Next, taking gradient of the $I$-equation and then multiplying it by $\nabla
I|\nabla I|^{2}$ and integrating by parts, we conclude that

$\begin{split}&\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
I|^{4}+d_{I}\int_{\Omega}\left|\nabla\left|\nabla
I\right|^{2}\right|^{2}+2d_{I}\int_{\Omega}\left|\nabla
I\right|^{2}\left|D^{2}I\right|^{2}+2(\gamma+\mu)\int_{\Omega}\left|\nabla
I\right|^{4}\\\ &=\int_{\partial\Omega}|\nabla
I|^{2}\frac{\partial}{\partial\nu}\left|\nabla
I\right|^{2}-2\beta\int_{\Omega}S^{q}I^{p}\Delta I|\nabla
I|^{2}-2\beta\int_{\Omega}S^{q}I^{p}\nabla I\cdot\nabla|\nabla
I|^{2}.\end{split}$ (3.51)

In the above, we have used the identity

$2\nabla I\cdot\nabla\Delta I=\Delta|\nabla I|^{2}-2\left|D^{2}I\right|^{2},$

where
$\left|D^{2}I\right|^{2}=\sum_{i,j=1}^{n}\left|\frac{\partial^{2}I}{\partial
x_{i}\partial x_{j}}\right|^{2}$. For the boundary integral in (3.51), we can
handle it (cf. [18, 48, 61, 62]) as follows

$\begin{split}\int_{\partial\Omega}\left|\nabla
I\right|^{2}\frac{\partial}{\partial\nu}\left|\nabla
I\right|^{2}&\leq\epsilon\int_{\Omega}\left|\nabla\left|\nabla
I\right|^{2}\right|^{2}+C_{\epsilon}\left(\int_{\Omega}|\nabla
I|^{2}\right)^{2}\\\ &\leq\epsilon\int_{\Omega}\left|\nabla\left|\nabla
I\right|^{2}\right|^{2}+C_{\epsilon},\quad\quad\forall\epsilon>0.\end{split}$
(3.52)

For the remaining terms in (3.51), we apply Hölder and Young’s inequality to
deduce that

$\begin{split}&-2\beta\int_{\Omega}S^{q}I^{p}\Delta I|\nabla
I|^{2}-2\beta\int_{\Omega}S^{q}I^{p}\nabla I\cdot\nabla\left|\nabla
I\right|^{2}\\\ &\leq\frac{2d_{I}}{n}\int_{\Omega}\left|\Delta
I\right|^{2}\left|\nabla
I\right|^{2}+\frac{n\beta^{2}}{2d_{I}}\|I\|_{L^{\infty}}^{2p}\int_{\Omega}S^{2q}\left|\nabla
I\right|^{2}\\\ &\ +\frac{d_{I}}{2}\int_{\Omega}\left|\nabla\left|\nabla
I\right|^{2}\right|^{2}+\frac{2\beta^{2}}{d_{I}}\|I\|_{L^{\infty}}^{2p}\int_{\Omega}S^{2q}\left|\nabla
I\right|^{2}\\\ &\leq 2d_{I}\int_{\Omega}\left|D^{2}I\right|^{2}\left|\nabla
I\right|^{2}+\frac{d_{I}}{2}\int_{\Omega}\left|\nabla\left|\nabla
I\right|^{2}\right|^{2}+\frac{(n+4)\beta^{2}}{2d_{I}}\|I\|_{L^{\infty}}^{2p}\int_{\Omega}S^{2q}\left|\nabla
I\right|^{2}.\end{split}$ (3.53)

Now, we claim, for $n=1,2$ and for any $\epsilon,\eta>0$, there exist $C_{2},\
C_{\epsilon,\eta}>0$ such that

$\int_{\Omega}S^{2q}|\nabla I|^{2}\leq C_{2}\epsilon\|\nabla|\nabla
I|^{2}\|_{L^{2}}^{2}+C_{2}\epsilon+\eta\|\nabla
S\|_{L^{2}}^{2}+C_{\epsilon,\eta}.$ (3.54)

We distinguish the case of $n=1$ and $n=2$ to prove this claim.

Case of $n=1$. Given the $(L^{1},L^{2})$-boundness of $(S,\nabla I)$ in
(3.36), upon twice applications of the 1-D G-N interpolation inequality, we
find that

$\left\\{\forall\eta>0,\ r\in(1,\infty],\
s<\frac{3r}{r-1}\right\\}\Longrightarrow\begin{cases}\|S\|_{L^{r}}^{s}\leq\eta\|\nabla
S\|_{L^{2}}^{2}+C_{\eta},\\\\[7.11317pt] \left\|\left|\nabla
I\right|^{2}\right\|_{L^{r}}^{\frac{3r}{r-1}}\leq
C_{3}\left\|\nabla\left|\nabla
I\right|^{2}\right\|_{L^{2}}^{2}+C_{3}.\end{cases}$ (3.55)

Recall that $q<\frac{3}{2}$ forced by (3.35), we then can fix $r\in(1,\infty)$
such that

$1<r<\frac{1}{(1-2q)^{+}}\Longleftrightarrow\left\\{\frac{2qr}{r-1}:=s>1,\ \
2q<\frac{6qr}{2r+1}<\frac{3s}{s-1}\right\\};$

with such chosen $r$, we deduce from (3.55), Hölder and Young’s inequality
with epsilon, for any $\epsilon,\eta>0$, that

$\displaystyle\int_{\Omega}S^{2q}\left|\nabla I\right|^{2}$
$\displaystyle\leq\left\|S\right\|_{L^{\frac{2qr}{r-1}}}^{2q}\left\|\left|\nabla
I\right|^{2}\right\|_{L^{r}}$ $\displaystyle\leq\epsilon\left\|\left|\nabla
I\right|^{2}\right\|_{L^{r}}^{\frac{3r}{r-1}}+C_{\epsilon}\left\|S\right\|_{L^{\frac{2qr}{r-1}}}^{\frac{6qr}{2r+1}}$
$\displaystyle\leq C_{4}\epsilon\left\|\nabla\left|\nabla
I\right|^{2}\right\|_{L^{2}}^{2}+C_{4}\epsilon+\eta\left\|\nabla
S\right\|_{L^{2}}^{2}+C_{\epsilon,\eta},$

yielding trivially (3.54) in the case of $n=1$. In particular, this implies

$\displaystyle\int_{\Omega}S^{2}\left|\nabla I\right|^{2}\leq
C_{4}\epsilon\left\|\nabla\left|\nabla
I\right|^{2}\right\|_{L^{2}}^{2}+C_{4}\epsilon+\eta\left\|\nabla
S\right\|_{L^{2}}^{2}+C_{\epsilon,\eta}.$ (3.56)

Case of $n=2$. The Young’s inequality with epsilon entails

$\int_{\Omega}S^{2q}|\nabla I|^{2}\leq\epsilon\int_{\Omega}|\nabla
I|^{6}+C_{\epsilon}\int_{\Omega}S^{3q},\quad\quad\forall\epsilon>0.$ (3.57)

In view of the boundedness of $\|\nabla I\|_{L^{2}}$ by (3.36), the 2-D G-N
inequality shows

$\begin{split}\int_{\Omega}\left|\nabla I\right|^{6}=\left\|\left|\nabla
I\right|^{2}\right\|_{L^{3}}^{3}&\leq C_{5}\left(\left\|\nabla\left|\nabla
I\right|^{2}\right\|_{L^{2}}^{2}\left\|\left|\nabla
I\right|^{2}\right\|_{L^{1}}+\left\|\left|\nabla
I\right|^{2}\right\|_{L^{1}}^{3}\right)\\\ &\leq
C_{6}\left\|\nabla\left|\nabla
I\right|^{2}\right\|_{L^{2}}^{2}+C_{6}.\end{split}$ (3.58)

Using the 2-D G-N inequality and noting that $q<1$ implied by (3.35), we can
easily infer

$\int_{\Omega}S^{3q}\leq\eta\int_{\Omega}|\nabla
S|^{2}+C_{\eta},\quad\quad\forall\eta>0.$ (3.59)

Substituting (3.58) and (3.59) into (3.57), we discover that (3.54) holds for
$n=2$ as well.

Thus, by suitably choosing $\epsilon>0$, we derive from (3.51), (3.52), (3.53)
and (3.54) that

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
I|^{4}+\frac{d_{I}}{3}\int_{\Omega}\left|\nabla\left|\nabla
I\right|^{2}\right|^{2}+2(\gamma+\mu)\int_{\Omega}\left|\nabla
I\right|^{4}\leq\eta\int_{\Omega}|\nabla S|^{2}+C_{\eta}.$ (3.60)

Now, we shall estimate the first item on the right-hand-side of (3.50). The
case of $n=1$ has been treated in (3.56). If $n=2$, instead of (3.59), it
holds

$\displaystyle\int_{\Omega}S^{3}\leq C_{7}\int_{\Omega}|\nabla S|^{2}+C_{7}.$
(3.61)

Then, in light of (3.61) and (3.58), for any $\epsilon>0$, similar to (3.57),
we have

$\displaystyle\int_{\Omega}S^{2}\left|\nabla I\right|^{2}$
$\displaystyle\leq\epsilon\int_{\Omega}S^{3}+C_{\epsilon}\int_{\Omega}\left|\nabla
I\right|^{6}$ $\displaystyle\leq C_{7}\epsilon\int_{\Omega}\left|\nabla
S\right|^{2}+C_{7}\epsilon+C_{\epsilon}\left\|\nabla\left|\nabla
I\right|^{2}\right\|_{L^{2}}^{2}+C_{7}.$ (3.62)

Recalling (3.56), we conclude that (3.62) still holds if $n=1$. Inserting this
inequality into (3.50) and choosing $\epsilon>0$ appropriately, we arrive at

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}S^{2}+\frac{d_{S}}{2}\int_{\Omega}|\nabla
S|^{2}+\beta\int_{\Omega}S^{1+q}I^{p}\leq
C_{8}\int_{\Omega}\left|\nabla\left|\nabla I\right|^{2}\right|+C_{8}.$ (3.63)

Multiplying (3.60) by $\frac{6C_{8}}{d_{I}}$ and combining (3.63) so that the
first term on the right-hand-side of (3.63) is annihilated and then choosing
$\eta>0$ with $\frac{d_{S}}{2}-\frac{6C_{8}}{d_{I}}\eta=\frac{d_{S}}{4}$, we
find

$\displaystyle\frac{d}{dt}\int_{\Omega}\left(S^{2}+\left|\nabla
I\right|^{4}\right)+C_{9}\int_{\Omega}\left(\left|\nabla
S\right|^{2}+S^{1+q}I^{p}+\left|\nabla I\right|^{4}+\left|\nabla\left|\nabla
I\right|^{2}\right|\right)\leq C_{10}.$ (3.64)

Finally, in light of the consequence of $n$-D G-N inequality, it follows
readily that

$\forall\,\epsilon>0,\ \exists\,C_{\epsilon}>0\ \text{ s.t.
}\int_{\Omega}S^{2}\leq\epsilon\int_{\Omega}|\nabla S|^{2}+C_{\epsilon},$

we derive from (3.64) a key differential inequality as follows:

$\displaystyle\frac{d}{dt}\int_{\Omega}\left(S^{2}+\left|\nabla
I\right|^{4}\right)+C_{11}\int_{\Omega}\left(S^{2}+\left|\nabla
S\right|^{2}+S^{1+q}I^{p}+\left|\nabla I\right|^{4}+\left|\nabla\left|\nabla
I\right|^{2}\right|\right)\leq C_{12}.$

which, upon being solved, yields readily (3.49). ∎

Armed with the global boundedness information provided by Lemmas 3.7 and 3.8,
we are now at the position to present the uniform boundedness of
$\|S\|_{L^{\infty}}$ and $\|I\|_{W^{1,\infty}}$ and thus the global existence
for (1.1) in dimensions one and two.

######  Theorem 3.9.

Let $\Omega\subset\mathbb{R}^{n}\ (n=1,2)$ be a bounded and smooth domain and,
let (3.35) hold. Suppose the initial data fulfills (1.2). Then the unique
classical solution of (1.1) exists globally in time and is uniformly bounded
in the sense of (3.12).

###### Proof.

First, we utilize the $(L^{2},L^{\infty})$-boundedness of $(S,I)$ in (3.49)
and (3.37), the $L^{r}$-$L^{s}$ type smoothing estimates in Lemma 2.2 and the
fact $nq<2$ implied by (3.35) to deduce from the variation-of-constants formal
for $I$ in (3.48) that, for all $t\in(0,T_{m})$,

$\begin{split}\|\nabla I(\cdot,t)\|_{L^{\infty}}&\leq\|\nabla
e^{t(d_{I}\Delta-(\gamma+\mu))}I_{0}\|_{L^{\infty}}+\beta\int_{0}^{t}\|\nabla
e^{(t-s)(d_{I}\Delta-(\gamma+\mu))}S^{q}I^{p}\|_{L^{\infty}}ds\\\ &\leq
C\|\nabla
I_{0}\|_{L^{\infty}}+C\int_{0}^{t}\left[1+(t-s)^{-\frac{1}{2}-\frac{nq}{4}}\right]e^{-(\gamma+\mu)(t-s)}\|S^{q}I^{p}\|_{L^{\frac{2}{q}}}ds\\\
&\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{2+nq}{4}}\right]e^{-(\gamma+\mu)(t-s)}\|S\|_{L^{2}}^{q}\|I\|_{L^{\infty}}^{p}ds\\\
&\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{2+nq}{4}}\right]e^{-(\gamma+\mu)(t-s)}ds\\\
&\leq
C+C\int_{0}^{\infty}\left(1+\tau^{-\frac{2+nq}{4}}\right)e^{-(\gamma+\mu)\tau}d\tau\leq
C.\end{split}$ (3.65)

Next, we use the variation-of-constants formula to the $S$-equation in (1.1)
to write $S$ as

$\begin{split}S(t)&=e^{t(d_{S}\Delta-1)}S_{0}+\chi\int_{0}^{t}e^{(t-s)(d_{S}\Delta-1)}\nabla\cdot(S\nabla
I)ds\\\ &\ \
-\beta\int_{0}^{t}e^{(t-s)(d_{S}\Delta-1)}S^{q}I^{p}ds+\int_{0}^{t}e^{(t-s)(d_{S}\Delta-1)}(\gamma
I+S)ds.\end{split}$ (3.66)

Case of $n=1$: Applying the smoothing estimates of the Neumann heat semigroup
on (3.66), using the obtained $(L^{2},W^{1,\infty})$-boundedness of $(S,I)$ in
(3.49), (3.37), (3.46) and the fact that $q<\frac{3}{2}$ by (3.35), we see,
for all $t\in(0,T_{m})$,

$\displaystyle\|S(\cdot,t)\|_{L^{\infty}}$
$\displaystyle\leq\left\|e^{t(d_{S}\Delta-1)}S_{0}\right\|_{L^{\infty}}+\chi\int_{0}^{t}\left\|e^{(t-s)(d_{S}\Delta-1)}\nabla\cdot\left(S\nabla
I\right)\right\|_{L^{\infty}}ds$ $\displaystyle\
\quad+\beta\int_{0}^{t}\left\|e^{(t-s)(d_{S}\Delta-1)}S^{q}I^{p}\right\|_{L^{\infty}}ds+\int_{0}^{t}\left\|e^{(t-s)(d_{S}\Delta-1)}\left(\gamma
I+S\right)\right\|_{L^{\infty}}ds$ $\displaystyle\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{1}{2}-\frac{1}{4}}\right]e^{-(t-s)}\left\|S\nabla
I\right\|_{L^{2}}ds$ $\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-\frac{q}{4}}\right]e^{-(t-s)}\left\|S^{q}I^{p}\right\|_{L^{\frac{2}{q}}}ds$
$\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-\frac{1}{4}}\right]e^{-(t-s)}\left\|\gamma
I+S\right\|_{L^{2}}ds$ $\displaystyle\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{3}{4}}\right]e^{-(t-s)}\|S\|_{L^{2}}\|\nabla
I\|_{L^{\infty}}ds$ $\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-\frac{q}{4}}\right]e^{-(t-s)}\|S\|_{L^{2}}^{q}\|I\|_{L^{\infty}}^{p}ds$
$\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-\frac{1}{4}}\right]e^{-(t-s)}\left(\|I\|_{L^{2}}+\|S\|_{L^{2}}\right)ds$
$\displaystyle\leq
C+C\int_{0}^{\infty}\left(1+\tau^{-\frac{3}{4}}+\tau^{-\frac{q}{4}}+\tau^{-\frac{1}{4}}\right)e^{-\tau}d\tau\leq
C.$ (3.67)

Case of $n=2$: Unlike Case of $n=1$, we need to further improve the
$L^{2}$-regularity of $S$ to its $L^{3}$-regularity. To this purpose, by the
$L^{r}$-$L^{s}$ type smoothing estimates of the Neumann heat semigroup, we use
the established $(L^{2},W^{1,\infty})$-boundedness of $(S,I)$ and the fact
$q<1$ implied by (3.35) to deduce first from (3.66) that

$\displaystyle\|S(\cdot,t)\|_{L^{3}}$
$\displaystyle\leq\left\|e^{t(d_{S}\Delta-1)}S_{0}\right\|_{L^{3}}+\chi\int_{0}^{t}\left\|e^{(t-s)(d_{S}\Delta-1)}\nabla\cdot(S\nabla
I)\right\|_{L^{3}}ds$ $\displaystyle\
\quad+\beta\int_{0}^{t}\left\|e^{(t-s)(d_{S}\Delta-1)}S^{q}I^{p}\right\|_{L^{3}}ds+\int_{0}^{t}\left\|e^{(t-s)(d_{S}\Delta-1)}\left(\gamma
I+S\right)\right\|_{L^{3}}ds$ $\displaystyle\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{1}{2}-\left(\frac{1}{2}-\frac{1}{3}\right)}\right]e^{-(t-s)}\left\|S\nabla
I\right\|_{L^{2}}ds$ $\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-\left(\frac{q}{2}-\frac{1}{3}\right)}\right]e^{-(t-s)}\left\|S^{q}I^{p}\right\|_{L^{\frac{2}{q}}}ds$
$\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-(\frac{1}{2}-\frac{1}{3})}\right]e^{-(t-s)}\left\|\gamma
I+S\right\|_{L^{2}}ds$ $\displaystyle\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{2}{3}}\right]e^{-(t-s)}\|S\|_{L^{2}}\|\nabla
I\|_{L^{\infty}}ds$ $\displaystyle\ \
+C\int_{0}^{t}\left[1+(t-s)^{-\frac{3q-2}{6}}\right]e^{-(t-s)}\|S\|_{L^{2}}^{q}\|I\|_{L^{\infty}}^{p}ds$
$\displaystyle\ \
+C\int_{0}^{t}\left[1+(t-s)^{-\frac{1}{6}}\right]e^{-(t-s)}\left(\|I\|_{L^{2}}+\|S\|_{L^{2}}\right)ds$
$\displaystyle\leq
C+C\int_{0}^{\infty}\left(1+\tau^{-\frac{2}{3}}+\tau^{-\frac{3q-2}{6}}+\tau^{-\frac{1}{6}}\right)e^{-\tau}d\tau$
$\displaystyle\leq C.$ (3.68)

Then, using the just obtained $(L^{3},W^{1,\infty})$-boundedness of $(S,I)$
and the fact $q<1$, similar to (3.68), we finally conclude that

$\displaystyle\|S(\cdot,t)\|_{L^{\infty}}$
$\displaystyle\leq\left\|e^{t(d_{S}\Delta-1)}S_{0}\right\|_{L^{\infty}}+\chi\int_{0}^{t}\left\|e^{(t-s)(d_{S}\Delta-1)}\nabla\cdot(S\nabla
I)\right\|_{L^{\infty}}ds$ $\displaystyle\
\quad+\beta\int_{0}^{t}\left\|e^{(t-s)(d_{S}\Delta-1)}S^{q}I^{p}\right\|_{L^{\infty}}ds+\int_{0}^{t}\left\|e^{(t-s)(d_{S}\Delta-1)}\left(\gamma
I+S\right)\right\|_{L^{\infty}}ds$ $\displaystyle\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{1}{2}-\frac{1}{3}}\right]e^{-(t-s)}\left\|S\nabla
I\right\|_{L^{3}}ds$ $\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-\frac{q}{3}}\right]e^{-(t-s)}\|S^{q}I^{p}\|_{L^{\frac{3}{q}}}ds$
$\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-\frac{1}{3}}\right]e^{-(t-s)}\left\|\gamma
I+S\right\|_{L^{3}}ds$ $\displaystyle\leq
C+C\int_{0}^{t}\left[1+(t-s)^{-\frac{5}{6}}\right]e^{-(t-s)}\|S\|_{L^{3}}\|\nabla
I\|_{L^{\infty}}ds$ $\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-\frac{q}{3}}\right]e^{-(t-s)}\|S\|_{L^{3}}^{q}\|I\|_{L^{\infty}}^{p}ds$
$\displaystyle\
\quad+C\int_{0}^{t}\left[1+(t-s)^{-\frac{1}{3}}\right]e^{-(t-s)}\left(\|I\|_{L^{3}}+\|S\|_{L^{3}}\right)ds$
$\displaystyle\leq
C+C\int_{0}^{\infty}\left(1+\tau^{-\frac{5}{6}}+\tau^{-\frac{q}{3}}+\tau^{-\frac{1}{3}}\right)e^{-\tau}d\tau$
$\displaystyle\leq C.$ (3.69)

Combining (3.37), (3.67), (3.69) and (3.5), we achieve our desired uniform-in-
time estimate (3.12) on $[0,T_{m})$. Consequently, the extensibility criterion
(2.2) in the local existence of Lemma 2.3 yields first that $T_{m}=\infty$ and
then the boundedness (3.12) for all $t>0$. ∎

##  4\. Long time behavior of global bounded solutions

For any bounded global classical solution $(S,I)$ of (1.1) obeying (3.12),
applying the Hölder estimates for parabolic equations (cf. [37, Theorem 1.3])
and then using the standard parabolic Schauder theory (cf. [14, 22])
repeatedly (see similar argument in [29]), we see there exist $\theta\in(0,1)$
and $C_{0}>0$ such that

$\|S\|_{C^{2+\theta,1+\frac{\theta}{2}}(\overline{\Omega}\times[t,t+1])}+\|I\|_{C^{2+\theta,1+\frac{\theta}{2}}(\overline{\Omega}\times[t,t+1])}\leq
C_{0},\quad\forall\ t\geq 1.$ (4.1)

###  4.1.  With cross-diffusion $\chi\neq 0$ and with motality $\mu>0$

Our main result in this subsection reads as follows.

######  Theorem 4.1.

In the case of $\mu>0$, the cross-diffusive SIS model (1.1) does not admit
threshold dynamics. More precisely, for any bounded global classical solution
$(S,I)$ of (1.1) obeying (3.12), it follows, as $t\rightarrow\infty$, that

$\left(S(\cdot,t),I(\cdot,t)\right)\rightarrow(S_{*},0)$ (4.2)

uniformly on $\overline{\Omega}$, where

$\begin{cases}S_{*}=0,&\text{ if }0<p<1,\\\\[5.69046pt]
0<S_{*}=|\Omega|^{-1}\left(\int_{\Omega}\left(S_{0}+I_{0}\right)-\mu\int_{0}^{\infty}\int_{\Omega}I\right),&\text{
if }p\geq 1.\end{cases}$ (4.3)

Furthermore, $S_{*}\leq\left(\frac{\gamma+\mu}{\beta}\right)^{\frac{1}{q}}$ if
$p=1$, and $(S,I)\rightarrow(S_{*},0)$ exponentially if $p>1$.

###### Proof.

An integration of (2.4) shows, for all $t>0$, that

$\int_{\Omega}\left(S+I\right)+\mu\int_{0}^{t}\int_{\Omega}I=\int_{\Omega}\left(S_{0}+I_{0}\right),$
(4.4)

which along with the fact $\mu>0$ implies that

$\int_{0}^{\infty}\int_{\Omega}I\leq\frac{1}{\mu}\int_{\Omega}\left(S_{0}+I_{0}\right)<\infty.$
(4.5)

By the regularity (4.1), the integrand $\int_{\Omega}I$ is uniformly bounded
and uniformly continuous, and so $\|I(\cdot,t)\|_{L^{1}}\rightarrow 0$ as
$t\rightarrow\infty$. This, combined with (4.1) and the standard embedding
theorem (cf. (4.12) below or [61, 62] for instance), yields that

$I(\cdot,t)\to 0{\rm\ uniformly\ on\ }\overline{\Omega},\quad{\rm as\
}t\to\infty.$ (4.6)

As a result, since $\int_{\Omega}(S+I)$ is decreasing and is nonnegative, we
can assume

$\bar{S}:=\frac{1}{|\Omega|}\int_{\Omega}S(\cdot,t)\to S_{*},\quad{\mbox{a}s\
}t\to\infty,$ (4.7)

for some nonnegative constant $S_{*}$.

Now, we claim that

$\displaystyle S(\cdot,t)\to S_{*}{\rm\ uniformly\ on\
}\overline{\Omega},\quad{\rm as\ }t\to\infty.$ (4.8)

To this aim, we notice from the $S$-equation in (1.1) and (4.1) that

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}S^{2}$
$\displaystyle=-d_{S}\int_{\Omega}|\nabla S|^{2}-\chi\int_{\Omega}S\nabla
I\cdot\nabla S-\beta\int_{\Omega}S^{q+1}I^{p}+\gamma\int_{\Omega}SI$
$\displaystyle\leq-\frac{d_{S}}{2}\int_{\Omega}|\nabla
S|^{2}+\frac{\chi^{2}}{2d_{S}}\int_{\Omega}S^{2}|\nabla
I|^{2}+\gamma\int_{\Omega}SI$
$\displaystyle\leq-\frac{d_{S}}{2}\int_{\Omega}|\nabla
S|^{2}+\frac{C_{0}^{2}\chi^{2}}{2d_{S}}\int_{\Omega}|\nabla I|^{2}+\gamma
C_{0}\int_{\Omega}I,\quad\forall\,t\geq 1.$ (4.9)

Similarly, one can readily obtain from the $I$-equation that

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}I^{2}$
$\displaystyle=-d_{I}\int_{\Omega}|\nabla
I|^{2}+\beta\int_{\Omega}S^{q}I^{p+1}-(\gamma+\mu)\int_{\Omega}I^{2}$
$\displaystyle\leq-d_{I}\int_{\Omega}|\nabla I|^{2}+\beta
C_{0}^{p+q}\int_{\Omega}I,\quad\forall\,t\geq 1.$

Upon an integration and a use of (4.5), we find that

$\displaystyle\int_{1}^{\infty}\int_{\Omega}|\nabla I|^{2}$
$\displaystyle\leq\frac{\beta
C_{0}^{p+q}}{d_{I}}\int_{1}^{\infty}\int_{\Omega}I+\frac{1}{2d_{I}}\int_{\Omega}I^{2}(\cdot,1)$
$\displaystyle\leq\frac{\beta C_{0}^{p+q}}{\mu
d_{I}}\int_{\Omega}\left(S_{0}+I_{0}\right)+\frac{1}{2d_{I}}C_{0}^{2}|\Omega|<\infty.$
(4.10)

Integrating (4.9) and using (4.5) and (4.10), we deduce, for some $C_{1}>0$
that

$\int_{1}^{\infty}\int_{\Omega}|\nabla S|^{2}\leq C_{1}<\infty.$

This along with the uniform continuity of $\int_{\Omega}|\nabla S|^{2}$ due to
(4.1) shows

$\displaystyle\int_{\Omega}|\nabla S(\cdot,t)|^{2}\to 0,\quad\mbox{as
}t\to\infty.$

Hence, we conclude from (4.7) and the Poincaré inequality that

$\begin{split}\int_{\Omega}\left|S(\cdot,t)-S_{*}\right|^{2}&=\left|\bar{S}-S_{*}\right|^{2}|\Omega|+\int_{\Omega}\left|S(\cdot,t)-\bar{S}\right|^{2}\\\
&\ \ \leq\left|\bar{S}-S_{*}\right|^{2}|\Omega|+C\int_{\Omega}\left|\nabla
S(\cdot,t)\right|^{2}\to 0,\ \ \mbox{as }t\to\infty,\end{split}$ (4.11)

which along with (4.1) and the G-N inequality in (2.1) shows

$\begin{split}\|S(\cdot,t)-S_{*}\|_{L^{\infty}}&\leq C\left(\|\nabla
S(\cdot,t)\|_{L^{\infty}}^{\frac{n}{n+2}}+\|S(\cdot,t)-S_{*}\|_{L^{2}}^{\frac{n}{n+2}}\right)\|S(\cdot,t)-S_{*}\|_{L^{2}}^{\frac{2}{n+2}}\cr&\
\leq C\|S(\cdot,t)-S_{*}\|_{L^{2}}^{\frac{2}{n+2}}\rightarrow 0\ \ \text{as}\
\ t\rightarrow\infty.\end{split}$ (4.12)

This precisely gives rise to (4.8).

(i) We now show that $S_{*}=0$ if $p<1$. Suppose on the contrary that
$S_{*}>0$. Then it follows from (4.6) and (4.8) there exists some $T_{0}>0$
such that

$\beta
S^{q}(x,t)-(\gamma+\mu)I^{1-p}(x,t)>0,\quad(x,t)\in\Omega\times(T_{0},\infty).$

As a result, we have

$\displaystyle I_{t}-d_{I}\Delta I=I^{p}\left[\beta
S^{q}-(\gamma+\mu)I^{1-p}\right]>0,\quad(x,t)\in\Omega\times(T_{0},\infty).$

A simple comparison argument then yields
$I(x,t)\geq\min_{x\in\overline{\Omega}}I(x,T_{0})>0$ for all
$(x,t)\in\overline{\Omega}\times[T_{0},\infty)$, contradicting (4.6). Thus, it
must hold $S_{*}=0$ if $p<1.$

(ii) In the sequel, we aim to prove that $S_{*}>0$, provided $p\geq 1$.
Suppose on the contrary that $S_{*}=0$. Then there exists some $T_{1}>1$
fulfilling

$\displaystyle\beta
S^{q}(x,t)I^{p-1}(x,t)-(\gamma+\mu)\leq-\frac{1}{2}(\gamma+\mu),\quad(x,t)\in\overline{\Omega}\times[T_{1},\infty).$
(4.13)

and

$\displaystyle\gamma-\beta
S^{q}(x,t)I^{p-1}(x,t)>0,\quad(x,t)\in\overline{\Omega}\times[T_{1},\infty).$
(4.14)

Thanks to (4.13), one can easily see from the $I$-equation in (1.1) that $I$
satisfies

$\left\\{\begin{array}[]{ll}\displaystyle I_{t}\leq d_{I}\Delta
I-\frac{1}{2}(\gamma+\mu)I,&x\in\Omega,\,t>T_{1},\\\ \vskip
4.0pt\cr\displaystyle\frac{\partial
I}{\partial\nu}=0,&x\in\partial\Omega,\,t>T_{1}.\end{array}\right.$

Consider the following corresponding ODE problem

$\left\\{\begin{array}[]{ll}\displaystyle\frac{dw}{dt}=-\frac{1}{2}(\gamma+\mu)w,&t>T_{1},\\\
\vskip 4.0pt\cr\displaystyle
w(T_{1})=\max_{x\in\overline{\Omega}}I(x,T_{1}).\end{array}\right.$

An application of the comparison principle yields

$\displaystyle 0<I(x,t)\leq
w(t)=\max_{x\in\overline{\Omega}}I(x,T_{1})e^{-\frac{1}{2}(\gamma+\mu)(t-T_{1})},\quad(x,t)\in\overline{\Omega}\times[T_{1},\infty).$

This yields with a convenient constant $C_{2}>0$ that

$\displaystyle 0<I(x,t)\leq
C_{2}e^{-\frac{1}{2}(\gamma+\mu)t},\quad(x,t)\in\overline{\Omega}\times[T_{1},\infty).$
(4.15)

Lemma 2.3 and the regularity (4.1) imply that
$f(x,t)=\beta\Delta\left(S^{q}I^{p}\right)\in
C^{0}(\overline{\Omega}\times[T_{1},\infty))$. Then, writing $z=\Delta I$, we
infer from the $I$-equation in (1.1) that

$\displaystyle\begin{cases}z_{t}=d_{I}\Delta
z-(\gamma+\mu)z+f(x,t),&(x,t)\in\Omega\times[T_{1},\infty),\\\ \vskip
4.0pt\cr\frac{\partial
z}{\partial\nu}=0,&(x,t)\in\partial\Omega\times[T_{1},\infty),\end{cases}$

which along with the standard Schauder estimate shows that
$\|z(\cdot,t)\|_{C^{2}}$ and thus $\|D^{3}I(\cdot,t)\|_{L^{\infty}}$ is
uniformly bounded due to the elliptic estimate.

In view of the exponential convergence of $I$ in (4.15), we make use of the
G-N inequality in Lemma 2.1 to obtain, for some $C_{4},\,C_{5}>0$, that

$\begin{split}\|D^{2}I(\cdot,t)\|_{L^{\infty}(\Omega)}&\leq
C_{3}\left(\|D^{3}I(\cdot,t)\|_{L^{\infty}(\Omega)}^{2/3}\|I(\cdot,t)\|_{L^{\infty}(\Omega)}^{1/3}+\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\right)\\\
&\leq C_{4}e^{-C_{5}t},\quad t\geq T_{1}.\end{split}$ (4.16)

In particular, this indicates that

$\displaystyle\Delta I(x,t)\geq-
C_{4}e^{-C_{5}t},\quad(x,t)\in\overline{\Omega}\times[T_{1},\infty).$ (4.17)

Now, observe from the $S$-equation in (1.1) that

$\displaystyle S_{t}$ $\displaystyle=d_{S}\Delta S+\chi\nabla I\cdot\nabla
S+\chi S\Delta I+I\left(\gamma-\beta S^{q}I^{p-1}\right)$ $\displaystyle\geq
d_{S}\Delta S+\chi\nabla I\cdot\nabla S+\chi S\Delta I$ $\displaystyle\geq
d_{S}\Delta S+\chi\nabla I\cdot\nabla S-C_{4}\chi e^{-C_{5}t}S,$

for $x\in\Omega$ and $t>T_{1}$, where we have used (4.14) and (4.17). Consider
the ODE problem

$\left\\{\begin{array}[]{ll}\displaystyle\frac{dv}{dt}=-C_{4}\chi
e^{-C_{5}t}v,&t>T_{1},\\\ \vskip 4.0pt\cr\displaystyle
v(T_{1})=\min_{x\in\overline{\Omega}}S(x,T_{1})>0.\end{array}\right.$

We again employ the comparison principle to derive

$\displaystyle S(x,t)\geq
v(t),\quad(x,t)\in\overline{\Omega}\times[T_{1},\infty).$

It can be easily verified, for $t>T_{1}$, that

$\displaystyle
v(t)=v(T_{1})\exp\left\\{\frac{C_{4}\chi}{C_{5}}e^{-C_{5}T_{1}}\left[e^{-C_{5}(t-T_{1})}-1\right]\right\\}\geq
v(T_{1})\exp\left\\{-\frac{C_{4}\chi}{C_{5}}e^{-C_{5}T_{1}}\right\\}>0.$

As a result, it holds

$\displaystyle\liminf_{t\to\infty}\min_{x\in\overline{\Omega}}S(x,t)\geq
v(T_{1})\exp\left\\{-\frac{C_{4}\chi}{C_{5}}e^{-C_{5}T_{1}}\right\\}>0,$

contradicting our assumption that $S_{*}=0$. Henceforth, we must have
$S_{*}>0$.

(iii) We shall illustrate that $S_{*}^{q}\leq\frac{\gamma+\mu}{\beta}$ if
$p=1$. We proceed on the contrary that $S_{*}^{q}>\frac{\gamma+\mu}{\beta}$ if
$p=1$. Then there must exist some $T_{2}>1$ such that $(\beta
S^{q}-\gamma-\mu)I>0$ on $\overline{\Omega}\times[T_{2},\infty)$. Thus, we
have

$\displaystyle I_{t}-d_{I}\Delta I>0,\quad x\in\Omega,\,t>T_{2}.$

Again, it follows easily from the comparison principle that
$I(x,t)\geq\min_{x\in\overline{\Omega}}I(x,T_{2})>0$ for all
$(x,t)\in\overline{\Omega}\times[T_{2},\infty)$. However, this is a
contradiction to (4.6).

Sending $t\rightarrow\infty$ in (4.4) and applying (4.6) and (4.8), we obtain
the implicit formula for $S_{*}$ in (4.3).

(iv) We shall show that the convergence rate of $\left(S,I\right)\to(S_{*},0)$
is exponential in the case of $p>1$. Since $I$ converges uniformly to zero,
the above arguments indeed have shown (cf. (4.13), (4.15) and (4.16)), for
some $C_{5},\,C_{6}>0$, that

$\|I(\cdot,t)\|_{W^{1,\infty}}\leq C_{5}e^{-C_{6}t},\ \ t\geq T_{1}.$ (4.18)

This along with (4.7), (4.4) and (4.3) allows us to deduce that

$\left|\overline{S}-S_{*}\right|=\frac{1}{|\Omega|}\left|\mu\int_{t}^{\infty}\int_{\Omega}I-\int_{\Omega}I\right|\leq
C_{7}e^{-C_{6}t},\ \ \ \ t\geq T_{1}.$ (4.19)

Now, testing the $S$-equation in (1.1) by $(S-S_{*})$ and using the
boundedness of $(S,I)$, we find that there exists some $C_{8}>0$ fulfilling

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}\left(S-S_{*}\right)^{2}$
$\displaystyle=-d_{S}\int_{\Omega}|\nabla S|^{2}-\chi\int_{\Omega}S\nabla
I\cdot\nabla S+\int_{\Omega}\left(\gamma-\beta
S^{q}I^{p-1}\right)\left(S-S_{*}\right)I$
$\displaystyle\leq-\frac{d_{S}}{2}\int_{\Omega}|\nabla S|^{2}+C_{8}\|\nabla
I(\cdot,t)\|_{L^{\infty}}^{2}+C_{8}\|I(\cdot,t)\|_{L^{\infty}},\quad\forall\,t\geq
T_{1},$

which along with (4.11), (4.18) and (4.19) enables us to derive a differential
inequality for $\|S-S_{*}\|_{L^{2}}^{2}$ as follows: for some
$a,\,\delta,\,C_{9}>0$ with $\delta<a$,

$\frac{d}{dt}\int_{\Omega}\left(S-S_{*}\right)^{2}+a\int_{\Omega}\left(S-S_{*}\right)^{2}\leq
C_{9}e^{-\delta t},\ \ t\geq T_{1}.$

This directly entails

$\int_{\Omega}\left(S(\cdot,t)-S_{*}\right)^{2}\leq
e^{-a(t-T_{1})}\int_{\Omega}\left(S(\cdot,T_{1})-S_{*}\right)^{2}+\frac{C_{9}}{a-\delta}e^{-\delta
t},\ \ t\geq T_{1},$

which, in conjunction with (4.12) and (4.18), yields our desired exponential
convergence of $(S,I)$ to $(S_{*},0)$. The proof is thus completed. ∎

###  4.2. Without cross-diffusion $\chi=0$ and without mortality $\mu=0$

In this subsection, we shall always assume that $\chi=0$ and $\mu=0$. That is,
we focus on the following reaction-diffusion SIS epidemic model

$\begin{cases}S_{t}=d_{S}\Delta S-\beta S^{q}I^{p}+\gamma
I,&x\in\Omega,\,t>0,\\\\[3.0pt] I_{t}=d_{I}\Delta I+\beta S^{q}I^{p}-\gamma
I,&x\in\Omega,\,t>0,\\\\[3.0pt] \frac{\partial S}{\partial\nu}=\frac{\partial
I}{\partial\nu}=0,&x\in\partial\Omega,\,t>0,\\\\[3.0pt] S(x,0)=S_{0}(x),\ \
I(x,0)=I_{0}(x),&x\in\Omega.\end{cases}$ (4.20)

From now on, we no longer assume that transmission rate $\beta$ and recovery
rate $\gamma$ and positive constants. Instead, we take spatial heterogeneity
into consideration. That is, we assume $\beta(x)$ and $\gamma(x)$ are positive
$C^{\alpha}(\overline{\Omega})$-smooth functions for some $\alpha>0$.
According to [40, Theorem 2.3], the model (4.20) admits a unique global and
bounded solution $(S,I)$ for any $p,\,q>0$ with $S(x,t),\,I(x,t)>0$ for all
$x\in\overline{\Omega}$ and $t>0$. As a result, (4.1) still holds. In
addition, it is easily seen that the total population number is conserved:

$\int_{\Omega}(S+I)=\int_{\Omega}(S_{0}+I_{0})=:N>0,\quad\forall\,t>0.$ (4.21)

Concerning the long-time behavior of solutions, in the case of $p\in(0,1)$,
the authors in [40] only showed the uniform persistence of solutions; if
$p=1$, a basic reproduction number $\mathcal{R}_{0}$ can be defined and the
uniform persistence property still holds if $\mathcal{R}_{0}>1$; if $p>1$,
their analysis about the long-time behavior of the ODE version of (4.20)
suggests that its dynamics depend on the initial data. To the best of our
knowledge, we are not aware of any further results on the global attractivity
of disease-free equilibrium (DFE) or endemic equilibrium (EE) for the
reaction-diffusion system (4.20) presented in literature, although there are
some studies [20, 23] on the S(E)IR type ODE and PDE models with varying total
population. Here, for $0<p\leq 1$, we shall investigate the global asymptotic
stability of DFE or EE in two special cases: (i) $\gamma(x)=r\beta(x)$ for
some constant $r>0$ and all $x\in\Omega$, and (ii) $d_{S}=d_{I}$.

We first consider the case of $0<p<1$.

Let $q,\,r>0$. For $S\in\left[0,\frac{N}{|\Omega|}\right]$, we define a
function

$f(S)=r\left(\frac{N}{|\Omega|}-S\right)^{1-p}-S^{q}.$

Then it can be readily checked that $f(0)>0$ and
$f\left(\frac{N}{|\Omega|}\right)<0$. Moreover, $f^{\prime}(S)<0$ for all
$S\in\left(0,\frac{N}{|\Omega|}\right)$. Consequently, there exists a unique
$S^{*}\in\left(0,\frac{N}{|\Omega|}\right)$ satisfying

$r\left(I^{*}\right)^{1-p}:=r\left(\frac{N}{|\Omega|}-S^{*}\right)^{1-p}=(S^{*})^{q}.$
(4.22)

Obviously, $(S^{*},I^{*})$ is an EE of (4.20) if
$\frac{\gamma(x)}{\beta(x)}\equiv r$ for all $x\in\Omega$.

######  Theorem 4.2.

Suppose $0<p<1$. Assume there exists some $r>0$ such that
$\gamma(x)=r\beta(x)$ for all $x\in\Omega$. Then, as $t\to\infty$, the unique
global and bounded solution $(S,I)$ of the reaction-diffusion model (4.20)
satisfies

$\left(S(x,t),I(x,t)\right)\to(S^{*},I^{*}),$

uniformly for $x\in\overline{\Omega}$, where $(S^{*},I^{*})$ is defined by
(4.22).

###### Proof.

Suppose $q\neq 1$. We define the following Lyapunov functional

$V_{1}(S,I)=\int_{\Omega}\left\\{\left[\left(S-S^{*}\right)-\frac{(S^{*})^{q}}{1-q}\left(S^{1-q}-(S^{*})^{1-q}\right)\right]+\left[(I-I^{*})-\frac{(I^{*})^{1-p}}{p}\left(I^{p}-(I^{*})^{p}\right)\right]\right\\}.$

Elementary analysis shows that $V_{1}(S,I)\geq 0$ for all $S,\,I>0$. Moreover,
in light of $(S^{*})^{q}=r(I^{*})^{1-p}$ due to (4.22), by direct
calculations, we compute from (4.20) that

$\displaystyle\frac{d}{dt}V_{1}(S,I)$
$\displaystyle=\int_{\Omega}\left[\left(1-\frac{(S^{*})^{q}}{S^{q}}\right)S_{t}+\left(1-\frac{(I^{*})^{1-p}}{I^{1-p}}\right)I_{t}\right]$
$\displaystyle=-qd_{S}(S^{*})^{q}\int_{\Omega}\frac{|\nabla
S|^{2}}{S^{q+1}}-d_{I}(1-p)(I^{*})^{1-p}\int_{\Omega}\frac{|\nabla
I|^{2}}{I^{2-p}}-(I^{*})^{1-p}\int_{\Omega}\beta\frac{(S^{q}I^{p}-rI)^{2}}{S^{q}I}$
$\displaystyle\leq 0.$

Thus, upon an integration in $t\in(1,\infty)$ and a use of the boundedness of
$(S,I)$, we find

$\int_{1}^{\infty}\int_{\Omega}\left[qrd_{S}\frac{|\nabla
S|^{2}}{S^{q+1}}+d_{I}(1-p)\frac{|\nabla
I|^{2}}{I^{2-p}}+\beta\frac{(S^{q}I^{p}-rI)^{2}}{S^{q}I}\right]<\infty.$
(4.23)

The regularity in (4.1) along with Ascoli-Arzelà theorem implies there exists
a sequence $t_{k}\rightarrow\infty$ and nonnegative $C^{2}$-function
$(S_{\infty},I_{\infty})$ such that
$(S(\cdot,t_{k}),I(\cdot,t_{k}))\rightarrow(S_{\infty},I_{\infty})$ in
$C^{2}(\overline{\Omega})$, as $k\rightarrow\infty$. One the other hand,
recall that $(S,I)$ is uniformly persistent by [40, Theorem 2.5], then the
function

$t\mapsto\int_{\Omega}\left[qrd_{S}\frac{|\nabla
S|^{2}}{S^{q+1}}+d_{I}(1-p)\frac{|\nabla
I|^{2}}{I^{2-p}}+\beta\frac{(S^{q}I^{p}-rI)^{2}}{S^{q}I}\right]$

is uniformly bounded and uniformly continuous. Thus, we infer from (4.23) that

$\int_{\Omega}\left[qrd_{S}\frac{|\nabla
S_{\infty}|^{2}}{S_{\infty}^{q+1}}+d_{I}(1-p)\frac{|\nabla
I_{\infty}|^{2}}{I_{\infty}^{2-p}}+\beta\frac{(S_{\infty}^{q}I_{\infty}^{p}-rI_{\infty})^{2}}{S_{\infty}^{q}I_{\infty}}\right]=0,$

which entails both $S_{\infty}$ and $I_{\infty}$ are constants and
$S^{q}_{\infty}I^{p}_{\infty}-rI_{\infty}=0$. In addition, $S_{\infty}$ and
$I_{\infty}$ must be positive constants again by the uniform persistence of
$(S,I)$. As a result, it must hold $(S_{\infty},I_{\infty})=(S^{*},I^{*})$ by
uniqueness. Since the limiting function $(S_{\infty},I_{\infty})$ is unique,
it follows that $(S(\cdot,t),I(\cdot,t))\to(S^{*},I^{*})$ in
$C^{2}(\overline{\Omega})$, as $t\to\infty$.

When $q=1$, we define the following Lyapunov functional

$V_{2}(S,I)=\int_{\Omega}\left\\{\left(S-S^{*}-S^{*}\ln\frac{S}{S^{*}}\right)+\left[(I-I^{*})-\frac{(I^{*})^{1-p}}{p}\left(I^{p}-(I^{*})^{p}\right)\right]\right\\}.$

Then $V_{2}(S,I)\geq 0$ for all $S,\,I>0$. Furthermore, we have

$\displaystyle\frac{d}{dt}V_{2}(S,I)$
$\displaystyle=\int_{\Omega}\left[\left(1-\frac{S^{*}}{S}\right)S_{t}+\left(1-\frac{(I^{*})^{1-p}}{I^{1-p}}\right)I_{t}\right]$
$\displaystyle=-d_{S}S^{*}\int_{\Omega}\frac{|\nabla
S|^{2}}{S^{2}}-d_{I}(1-p)(I^{*})^{1-p}\int_{\Omega}\frac{|\nabla
I|^{2}}{I^{2-p}}-(I^{*})^{1-p}\int_{\Omega}\beta\frac{(SI^{p}-rI)^{2}}{S^{q}I}$
$\displaystyle\leq 0.$

Then the rest of the argument is the same as that of $q\neq 1$ as done before.
∎

Now, suppose that $d_{S}=d_{I}$. For our later purpose, given constant
$\tau_{0}>0$, we consider the following initial-boundary value problem:

$\left\\{\begin{array}[]{lll}u_{t}=d_{I}\Delta
u+\beta(x)\left(\tau_{0}-u\right)^{q}u^{p}-\gamma(x)u,&x\in\Omega,\,t>0,\\\
\vskip 3.0pt\cr\frac{\partial u}{\partial\nu}=0,&x\in\partial\Omega,\,t>0,\\\
\vskip 3.0pt\cr u(x,0)=u_{0}(x),&x\in\Omega.\end{array}\right.$ (4.24)

######  Lemma 4.3.

Let $0<p<1$. Inside the interval $\left[0,\tau_{0}\right]$, the problem (4.24)
admits a unique positive steady state $U(x)$, which is globally asymptotically
stable for (4.24) with initial data fulfilling $\inf_{\Omega}u_{0}>0$ and
$\|u_{0}\|_{L^{\infty}}<\tau_{0}$.

###### Proof.

Since $0<p<1$, define

$\delta_{0}=\sup\left\\{\tilde{\delta}\in(0,\tau_{0}):\
\left(\tau_{0}-\delta\right)^{q}\inf_{\Omega}\frac{\beta}{\gamma}-\delta^{1-p}>0,\
\forall\delta\in(0,\tilde{\delta})\right\\}.$

Then one can easily check for any $\delta\in(0,\delta_{0}]$, that the two
positive constants $\delta$ and $\tau_{0}$, respectively, are lower and upper
solutions for the steady state problem of (4.24). Thus, the existence of a
positive equilibrium $U(x)\in\left[0,\tau_{0}\right]$ is guaranteed by the
standard upper-lower solutions method.

Next, we first show there exists a unique steady state in
$\left[\delta,\tau_{0}\right]$. To this end, we let $u_{1}$ and $u_{2}$ be the
minimal and maximal solutions in $\left[\delta,\tau_{0}\right]$. It then
follows that $\delta\leq u_{1}\leq u_{2}\leq\tau_{0}$ on $\overline{\Omega}$.
Then we multiply by $u_{2}$ the equation satisfied by $u_{1}$ to derive

$-d_{I}\int_{\Omega}\nabla u_{1}\cdot\nabla
u_{2}+\int_{\Omega}\left[\beta\left(\tau_{0}-u_{1}\right)^{q}u_{1}^{p}u_{2}-\gamma
u_{1}u_{2}\right]=0.$ (4.25)

By interchanging the role of $u_{1}$ and $u_{2}$, we have

$-d_{I}\int_{\Omega}\nabla u_{1}\cdot\nabla
u_{2}+\int_{\Omega}\left[\beta\left(\tau_{0}-u_{2}\right)^{q}u_{2}^{p}u_{1}-\gamma
u_{1}u_{2}\right]=0.$ (4.26)

Thus, we see from (4.25) and (4.26) that

$\int_{\Omega}\beta
u_{1}^{p}u_{2}^{p}\left[\left(\tau_{0}-u_{1}\right)^{q}u_{2}^{1-p}-\left(\tau_{0}-u_{2}\right)^{q}u_{1}^{1-p}\right]=0.$

This along with the facts
$\left(\tau_{0}-u_{1}\right)^{q}\geq\left(\tau_{0}-u_{2}\right)^{q}>0$ and
$u_{2}^{1-p}\geq u_{1}^{1-p}$ shows that $u_{1}\equiv u_{2}$ on
$\overline{\Omega}$, yielding the desired uniqueness on $[\delta,\tau_{0}]$.
Now, for any two positive equilibria $u_{1}$ and $u_{2}$ on $[0,\tau_{0}]$, we
put $\delta=\min\\{\min_{\overline{\Omega}}u_{1},\
\min_{\overline{\Omega}}u_{2},\ \delta_{0}\\}$; then $\delta>0$ and
$u_{1},u_{2}$ are two equilibria on $[\delta,\tau_{0}]$, and so the proved
uniqueness shows $u_{1}\equiv u_{2}$ on $\overline{\Omega}$.

Given initial data fulfilling $\inf_{\Omega}u_{0}>0$ and
$\|u_{0}\|_{L^{\infty}}<\tau_{0}$, for any
$0<\delta\leq\min\\{\inf_{\Omega}u_{0},\ \delta_{0}\\}$, we see that
$\delta\leq u_{0}(x)\leq\tau_{0}$ for all $x\in\overline{\Omega}$. Denoting by
$u(x,t;\delta)$ and $u(x,t;\tau_{0})$, respectively, the solutions of (4.24)
with initial data $u(x,0)\equiv\delta$ and $u(x,0)\equiv\tau_{0}$, we then
conclude from the maximum principle that

$\delta\leq u(x,t;\delta)\leq u(x,t)\leq
u\left(x,t;\tau_{0}\right)\leq\tau_{0},$ (4.27)

for all $(x,t)\in\overline{\Omega}\times(0,\infty)$. In addition,
$u(x,t;\delta)$ is non-decreasing and $u(x,t;\tau_{0})$ is non-increasing with
respect to $t\in(0,\infty)$. As a result, the point-wise limits

$\delta\leq\lim_{t\to\infty}u(x,t;\delta)=:\underline{U}(x)\quad\mbox{and}\quad\lim_{t\to\infty}u\left(x,t;\tau_{0}\right)=:\overline{U}(x)\leq\tau_{0},$
(4.28)

exist. Furthermore, notice that both $\underline{U}$ and $\overline{U}$ are
positive equilibria of (4.24) in $(0,\tau_{0})$, and thus,
$\underline{U}\equiv\overline{U}$. Hence, in light of the standard parabolic
regularity as in (4.1) or simply Dini’s theorem, we infer from (4.27) and
(4.28) that $u(x,t;u_{0})\rightarrow U(x)\equiv\underline{U}(x)$ as
$t\rightarrow\infty$ uniformly for $x\in\overline{\Omega}$. ∎

Now, we are ready to show the global asymptotic stability of EE when
$d_{S}=d_{I}$.

######  Theorem 4.4.

Let $0<p<1$ and $d_{S}=d_{I}$. Then, as $t\to\infty$, the unique global and
bounded solution $(S,I)$ of the reaction-diffusion model (4.20) fulfills

$(S(x,t),I(x,t))\to(\tilde{S}(x),\tilde{I}(x)),$

uniformly for $x\in\overline{\Omega}$, where $(\tilde{S}(x),\tilde{I}(x))$ is
the unique endemic equilibrium of (4.20).

###### Proof.

Let $w(x,t)=S(x,t)+I(x,t)$. Obviously, $w$ satisfies

$\left\\{\begin{array}[]{llll}w_{t}=d_{S}\Delta w,&x\in\Omega,\,t>0,\\\ \vskip
4.0pt\cr\frac{\partial w}{\partial\nu}=0,&x\in\partial\Omega,\,t>0,\\\ \vskip
4.0pt\cr w(x,0)=S_{0}(x)+I_{0}(x),&x\in\Omega,\\\ \vskip
4.0pt\cr\int_{\Omega}w=N,&t>0.\end{array}\right.$

Therefore, as $t\to\infty$, $w(x,t)\to\frac{N}{|\Omega|}$ uniformly for
$x\in\overline{\Omega}$. So, for any small $\varepsilon>0$, there exists some
$T=T(\varepsilon)>0$ such that

$\frac{N}{|\Omega|}-I(x,t)-\varepsilon<S(x,t)<\frac{N}{|\Omega|}-I(x,t)+\varepsilon,$
(4.29)

for all $x\in\overline{\Omega}$ and $t\geq T$. Consider the auxiliary problems

$\left\\{\begin{array}[]{lll}\bar{I}_{t}-d_{I}\Delta\bar{I}=\beta\left(\frac{N}{|\Omega|}+\varepsilon-\bar{I}\right)^{q}\bar{I}^{p}-\gamma\bar{I},&x\in\Omega,\,t>T,\\\
\vskip
3.0pt\cr\frac{\partial\bar{I}}{\partial\nu}=0,&x\in\partial\Omega,\,t>T,\\\
\vskip
3.0pt\cr\bar{I}(x,T)=\max_{x\in\overline{\Omega}}I(x,T),&x\in\Omega,\end{array}\right.$
(4.30)

and

$\left\\{\begin{array}[]{lll}\underline{I}_{t}-d_{I}\Delta\underline{I}=\beta\left(\frac{N}{|\Omega|}-\varepsilon-\underline{I}\right)^{q}\underline{I}^{p}-\gamma\underline{I},&x\in\Omega,\,t>T,\\\
\vskip
3.0pt\cr\frac{\partial\underline{I}}{\partial\nu}=0,&x\in\partial\Omega,\,t>T,\\\
\vskip
3.0pt\cr\underline{I}(x,T)=\min_{x\in\overline{\Omega}}I(x,T),&x\in\Omega,\end{array}\right.$
(4.31)

In view of the maximum principle for parabolic equations, comparing (4.20),
(4.30) and (4.31), we find that

$\underline{I}(x,t)\leq
I(x,t)\leq\bar{I}(x,t),\quad(x,t)\in\overline{\Omega}\times(T,\infty).$ (4.32)

It follows readily from (4.29) and the fact $S>0$ that

$0<\bar{I}(x,T)=\max_{x\in\overline{\Omega}}I(x,T)<\frac{N}{|\Omega|}+\varepsilon.$

In order to verify that the initial data in (4.31) fulfills the requirement in
Lemma 4.3 with $\tau_{0}=\frac{N}{|\Omega|}-\varepsilon$, let us recall from
[40, Theorem 2.5] that there exists some positive constant $\varepsilon_{0}$
depending only on $N$ such that
$\liminf_{t\to\infty}S(x,t)\geq\varepsilon_{0}$ uniformly for
$\overline{\Omega}$. Thus, by choosing $\varepsilon\in(0,\varepsilon_{0})$
small and enlarging $T$ if necessary, this, together with (4.29), yields

$0<\underline{I}(x,T)=\min_{x\in\overline{\Omega}}I(x,T)<\frac{N}{|\Omega|}-\varepsilon.$

Thus, we now apply Lemma 4.3 to problems (4.30) and (4.31) with
$\tau_{0}=\frac{N}{|\Omega|}+\varepsilon$ and
$\tau_{0}=\frac{N}{|\Omega|}-\varepsilon$, respectively, to conclude the
following uniform convergence

$\lim_{t\to\infty}\underline{I}(x,t)=U_{\varepsilon}(x)\quad\mbox{and}\quad\lim_{t\to\infty}\bar{I}(x,t)=U^{\varepsilon}(x),$
(4.33)

where $U^{\varepsilon}$ and $U_{\varepsilon}$ are the unique positive steady
state of (4.30) and (4.31), respectively.

Recall also from [40, Theorem 2.5] that there exists $\epsilon_{1}>0$ such
that $\liminf_{t\to\infty}I(x,t)\geq\varepsilon_{1}$ uniformly for
$\overline{\Omega}$. This allows one to infer there exists some small
$\delta>0$ such that $\left(\delta,\frac{N}{|\Omega|}-\delta\right)$ is a pair
of lower-and-upper solutions to the steady state problem of (4.30) for any
small $\varepsilon>0$. Hence, $0<\delta\leq
U^{\varepsilon}\leq\frac{N}{|\Omega|}-\delta$ for all $x\in\overline{\Omega}$
by the uniqueness of positive equilibrium recorded in Lemma 4.3. Then the
elliptic regularity estimates yield that $\\{U^{\varepsilon}\\}$ is bounded in
$C^{2+\alpha}(\overline{\Omega})$ for some $\alpha>0$. We then infer from the
Ascoli-Arzelà theorem that, along a subsequence of $\varepsilon\to 0$, it
holds $U^{\varepsilon}\to U^{0}\geq 0$ in $C^{2}(\overline{\Omega})$. By
sending $\varepsilon\to 0+$, we see that $U^{0}$ is in fact a positive
solution of the elliptic problem

$\left\\{\begin{array}[]{ll}-d_{I}\Delta
u=\beta\left(\frac{N}{|\Omega|}-u\right)^{q}u^{p}-\gamma u,&x\in\Omega,\\\
\vskip 3.0pt\cr\frac{\partial
u}{\partial\nu}=0,&x\in\partial\Omega.\end{array}\right.$ (4.34)

In the similar fashion, we have $\lim_{\varepsilon\to
0}U_{\varepsilon}=U_{0}>0$ in $C^{2}(\overline{\Omega})$, where $U_{0}$ is
also a positive solution of (4.34). On the other hand, as in the proof of
Lemma 4.3, one can show that the positive solution of (4.34) in
$(0,\frac{N}{|\Omega|}]$ is unique. Then it must hold $U_{0}\equiv U^{0}$ on
$\overline{\Omega}$.

Finally, by sending $\varepsilon\to 0+$, we conclude from (4.29), (4.32) and
(4.33) that

$\lim_{t\to\infty}(S(x,t),I(x,t))=\left(\frac{N}{|\Omega|}-\tilde{I}(x),\tilde{I}(x)\right)$

uniformly for $x\in\Omega$, where $\tilde{I}\equiv U_{0}\equiv U^{0}$ is the
unique positive solution of (4.34). Defining
$\tilde{S}(x)=\frac{N}{|\Omega|}-\tilde{I}(x)>0$ and noting $d_{S}=d_{I}$, one
can readily verify that $(\tilde{S},\tilde{I})$ is the unique endemic
equilibrium of (4.20). The proof is thus completed. ∎

Now, we consider the case of $p=1$. System (4.20) has a unique disease-free-
equilibrium (DFE) $(\frac{N}{|\Omega|},0)$. According to [50], we define the
basic reproduction number $\mathcal{R}_{0}$ via

$\mathcal{R}_{0}=\left(\frac{N}{|\Omega|}\right)^{q}\sup_{0\neq\varphi\in
H^{1}(\Omega)}\frac{\int_{\Omega}\beta\varphi^{2}}{\int_{\Omega}\left(d_{I}|\nabla\varphi|^{2}+\gamma\varphi^{2}\right)}.$
(4.35)

We here point out that $\mathcal{R}_{0}<1$ when $\lambda_{*}>0$,
$\mathcal{R}_{0}=1$ when $\lambda_{*}=0$ and $\mathcal{R}_{0}>1$ when
$\lambda_{*}<0$. Here and below, $\lambda_{*}$ is the principal eigenvalue of
the eigenvalue problem

$\left\\{\begin{array}[]{ll}-d_{I}\Delta\phi+\left(\gamma-\beta\left(\frac{N}{|\Omega|}\right)^{q}\right)\phi=\lambda\phi,&x\in\Omega,\\\
\vskip
3.0pt\cr\frac{\partial\phi}{\partial\nu}=0,&x\in\partial\Omega.\end{array}\right.$
(4.36)

The DFE $(\frac{N}{|\Omega|},0)$ is locally asymptotically stable if
$\mathcal{R}_{0}<1$, while it is unstable if $\mathcal{R}_{0}>1.$

Now, suppose there exists some $r>0$ such that $\gamma(x)=r\beta(x)$ for all
$x\in\overline{\Omega}$. It is not hard to see from (4.35) that
$\mathcal{R}_{0}=\frac{1}{r}\left(\frac{N}{|\Omega|}\right)^{q}$. If
$\mathcal{R}_{0}>1$, a constant EE
$(\hat{S},\hat{I}):=(r^{\frac{1}{q}},\frac{N}{|\Omega|}-r^{\frac{1}{q}})$
exists.

Concerning the large-time dynamics of (4.20), we have the following result.

######  Theorem 4.5.

Let $p=1$ and let $r>0$ be such that $\gamma(x)=r\beta(x)$ for all
$x\in\overline{\Omega}$. Then the reaction-diffusion SIS model (4.20) admits
threshold dynamics. More precisely, the unique classical solution $(S,I)$ of
(4.20) enjoys the following convergence properties:

  * (i)

If $\mathcal{R}_{0}\leq 1$, then as $t\to\infty$, the following uniform
convergence on $\overline{\Omega}$ holds:

$\left(S(\cdot,t),I(\cdot,t)\right)\to\left(\frac{N}{|\Omega|},0\right).$
(4.37)

  * (ii)

If $\mathcal{R}_{0}>1$, then as $t\to\infty$, the following uniform
convergence on $\overline{\Omega}$ holds:

$\left(S(\cdot,t),I(\cdot,t)\right)\to\left(r^{\frac{1}{q}},\frac{N}{|\Omega|}-r^{\frac{1}{q}}\right).$
(4.38)

###### Proof.

(i) Since $\mathcal{R}_{0}\leq 1$ implies
$r^{\frac{1}{q}}\geq\frac{N}{|\Omega|}$, we define the Lyapunov functional

$V_{3}(S,I)=\int_{\Omega}\left[\frac{1}{2}\left(S-\frac{N}{|\Omega|}\right)^{2}+\left(r^{\frac{1}{q}}-\frac{N}{|\Omega|}\right)I\right].$

Then we compute from (4.20) that

$\displaystyle\frac{d}{dt}V_{3}(S,I)$
$\displaystyle=\int_{\Omega}\left(S-\frac{N}{|\Omega|}\right)(d_{S}\Delta
S-\beta S^{q}I+r\beta I)$ $\displaystyle\
+\left(r^{\frac{1}{q}}-\frac{N}{|\Omega|}\right)\int_{\Omega}(d_{I}\Delta
I+\beta S^{q}I-r\beta I)$ $\displaystyle=-d_{S}\int_{\Omega}|\nabla
S|^{2}-\int_{\Omega}\beta
I\left(S^{q}-r\right)\left(S-r^{\frac{1}{q}}\right)\leq 0,$

which upon integration and the boundedness of $(S,I)$ implies

$\int_{1}^{\infty}\int_{\Omega}\left[d_{S}|\nabla S|^{2}+\beta
I\left(S^{q}-r\right)\left(S-r^{\frac{1}{q}}\right)\right]<\infty.$

The regularity in (4.1) along with Ascoli-Arzelà theorem implies there exists
$t_{k}\rightarrow\infty$ and nonnegative $C^{2}$-function
$(S_{\infty},I_{\infty})$ such that
$(S(\cdot,t_{k}),I(\cdot,t_{k}))\rightarrow(S_{\infty},I_{\infty})$ as
$k\rightarrow\infty$ in $C^{2}(\overline{\Omega})$. On the other hand, since
the function

$t\mapsto\int_{\Omega}\left[d_{S}|\nabla S|^{2}+\beta
I\left(S^{q}-r\right)\left(S-r^{\frac{1}{q}}\right)\right]$

is uniformly bounded and uniformly continuous, we infer that

$\int_{\Omega}|\nabla S_{\infty}|^{2}+\int_{\Omega}\beta
I_{\infty}\left(S_{\infty}^{q}-r\right)\left(S_{\infty}-r^{\frac{1}{q}}\right)=0,$
(4.39)

which entails either $S_{\infty}\equiv r^{1/q}$ or $I_{\infty}\equiv 0$. If
$S_{\infty}\equiv r^{1/q}$, then we infer from the conservation of total
population (4.21) that $I_{\infty}=\frac{N}{|\Omega|}-r^{\frac{1}{q}}$, which
is non-positive since $\mathcal{R}_{0}\leq 1$. Thus, we must have
$I_{\infty}\equiv 0$ and consequently $S_{\infty}\equiv\frac{N}{|\Omega|}$.
Since this type of argument can be applied to every subsequence of
$\\{t\\}_{t\geq 1}$, (4.37) follows readily.

(ii) Recall that
$(\hat{S},\hat{I}):=(r^{\frac{1}{q}},\frac{N}{|\Omega|}-r^{\frac{1}{q}})$ is
an EE. Define

$V_{4}(S,I)=\int_{\Omega}\left[\frac{1}{2}\left(S-\hat{S}+I-\hat{I}\right)^{2}+\frac{(d_{S}+d_{I})^{2}}{8d_{S}d_{I}}\left(S-\hat{S}\right)^{2}\right].$

Then

$\displaystyle\frac{d}{dt}V_{4}(S,I)$
$\displaystyle=\int_{\Omega}\left(S-\hat{S}+I-\hat{I}\right)\left(d_{S}\Delta
S+d_{I}\Delta I\right)$ $\displaystyle\quad\
+\frac{(d_{S}+d_{I})^{2}}{4d_{S}d_{I}}\int_{\Omega}\left(S-\hat{S}\right)\left(d_{S}\Delta
S-\beta S^{q}I+r\beta I\right)$ $\displaystyle=-d_{S}\int_{\Omega}|\nabla
S|^{2}-\int_{\Omega}\left|\frac{d_{S}+d_{I}}{2\sqrt{d_{I}}}\nabla
S+\sqrt{d_{I}}\nabla I\right|^{2}$ $\displaystyle\quad\
-\frac{(d_{S}+d_{I})^{2}}{4d_{S}d_{I}}\int_{\Omega}\beta
I\left(S-\hat{S}\right)\left(S^{q}-\hat{S}^{q}\right)\leq 0,$

which implies that (4.39) holds for some limiting $C^{2}$-nonnegative
functions $(S_{\infty},I_{\infty})$ of $(S,I)$. Hence, as argued above, we see
that

$(S_{\infty},I_{\infty})\in\left\\{\left(\frac{N}{|\Omega|},0\right),\
\left(\hat{S},\hat{I}\right)\right\\}.$ (4.40)

(Note that the limiting functions must be unique since
$\lim_{t\to\infty}V_{4}(S,I)$ exists and $V_{4}(\frac{N}{|\Omega|},0)\neq
V_{4}(\hat{S},\hat{I})$.) Since $\mathcal{R}_{0}>1$, the former case can be
eliminated by the uniform persistence property of $(S,I)$ in [40, Theorem
2.5], which is obtained by abstract dynamical system theory. Nevertheless,
here we would like to present an elementary argument to rule out this
possibility. In fact, suppose
$(S_{\infty},I_{\infty})=(\frac{N}{|\Omega|},0)$. Observe
$S_{\infty}=\frac{N}{|\Omega|}>r^{\frac{1}{q}}$ since $\mathcal{R}_{0}>1$.
Then there exists some positive constant $\varepsilon_{0}$ and $T>0$ such that

$S^{q}-r\geq\varepsilon_{0}>0\ \ \text{on}\ \
\overline{\Omega}\times[T,\infty).$

Hence, by the $I$-equation in (4.20), we find that

$\begin{cases}I_{t}\geq d_{I}\Delta
I+\varepsilon_{0}\left(\min_{\overline{\Omega}}\beta\right)I,&x\in\Omega,\,t>T,\\\\[3.0pt]
\frac{\partial I}{\partial\nu}=0,&x\in\partial\Omega,\,t>T,\\\\[3.0pt]
I(x,T)\geq\min_{x\in\overline{\Omega}}I(x,T),&x\in\Omega,\end{cases}$

which upon comparison with the corresponding ODE shows, for
$(x,t)\in\overline{\Omega}\times[T,\infty)$, that

$I(x,t)\geq\min_{x\in\overline{\Omega}}I(x,T)\exp\left\\{\varepsilon_{0}\left(\min_{\overline{\Omega}}\beta\right)\left(t-T\right)\right\\}.$

This clearly is incompatible with the boundedness of $I$. Then the desired
uniform convergence in (4.38) follows from (4.40). ∎

Finally, we turn our attention to the scenario of equal diffusion rates.

######  Theorem 4.6.

Suppose $p=1$ and $d_{S}=d_{I}$. Then the reaction-diffusion SIS model (4.20)
has the following threshold dynamics:

  * (i)

If $\mathcal{R}_{0}\leq 1$, then as $t\to\infty$, the following uniform
convergence on $\overline{\Omega}$ holds:

$\left(S(\cdot,t),I(\cdot,t)\right)\to\left(\frac{N}{|\Omega|},0\right).$
(4.41)

  * (ii)

If $\mathcal{R}_{0}>1$, then as $t\to\infty$, the following uniform
convergence on $\overline{\Omega}$ holds:

$\left(S(\cdot,t),I(\cdot,t)\right)\to\left(\tilde{S},\tilde{I}\right),$

where $(\tilde{S},\tilde{I})$ is the unique EE of (4.20).

###### Proof.

(i) Firstly, suppose $\mathcal{R}_{0}<1$. Then $\lambda_{*}>0$, where
$\lambda_{*}$ is the principal eigenvalue of (4.36). As in the reasoning in
Theorem 4.4, we are led to the auxliary problems (4.30) and (4.31) (with
$p=1$). To analyze the large-time dynamics of the unique solution to (4.30),
for any small $\varepsilon>0$, we consider the eigenvalue problem

$\left\\{\begin{array}[]{ll}-d_{I}\Delta\phi+\left(\gamma-\beta\left(\frac{N}{|\Omega|}+\varepsilon\right)^{q}\right)\phi=\lambda\phi,&x\in\Omega,\\\
\vskip
3.0pt\cr\frac{\partial\phi}{\partial\nu}=0,&x\in\partial\Omega,\end{array}\right.$

whose principal eigenvalue is denoted by $\lambda_{\varepsilon}$. It follows
from $\lambda_{*}>0$ that $\lambda_{\varepsilon}>0$ for small $\varepsilon>0$.
We then claim that (4.30) does not admit non-negative non-trivial steady state
in the interval $[0,\frac{N}{|\Omega|}+\varepsilon]$. In fact, suppose
$I_{\varepsilon}\geq 0,\,\not\equiv 0$ is such a steady state. Then we derive
from the variational formulation of $\lambda_{\varepsilon}$ and the equation
satisfied by $I_{\varepsilon}$ that

$\begin{split}\lambda_{\varepsilon}&=\inf_{0\neq\phi\in
H^{1}(\Omega)}\frac{\int_{\Omega}\left[d_{I}|\nabla\phi^{2}|+\left(\gamma-\beta\left(\frac{N}{|\Omega|}+\varepsilon\right)^{q}\right)\phi^{2}\right]}{\int_{\Omega}\phi^{2}}\\\
&\leq\frac{\int_{\Omega}\left[d_{I}|\nabla
I_{\varepsilon}^{2}|+\left(\gamma-\beta\left(\frac{N}{|\Omega|}+\varepsilon\right)^{q}\right)I_{\varepsilon}^{2}\right]}{\int_{\Omega}I_{\varepsilon}^{2}}\\\
&=\frac{\int_{\Omega}\beta
I_{\varepsilon}^{2}\left[\left(\frac{N}{|\Omega|}+\varepsilon-
I_{\varepsilon}\right)^{q}-\left(\frac{N}{|\Omega|}+\varepsilon\right)^{q}\right]}{\int_{\Omega}I_{\varepsilon}^{2}}\\\
&<0,\end{split}$ (4.42)

contradicting $\lambda_{\varepsilon}>0$. As a result,
$\lim_{t\to\infty}\bar{I}(\cdot,t)=0$ uniformly on $\overline{\Omega}$. Then
we derive from (4.32) that $\lim_{t\to\infty}I(\cdot,t)=0$ and therefore
$\lim_{t\to\infty}S(\cdot,t)=\frac{N}{|\Omega|}$ uniformly on
$\overline{\Omega}$.

Now suppose $\mathcal{R}_{0}=1$ or equivalently $\lambda_{*}=0$. This coupled
with (4.42) shows $\lambda_{\varepsilon}<0$ for small $\varepsilon>0$. Let
$\phi_{\varepsilon}>0$ be the corresponding principal eigenfunction,
normalized by $\|\phi_{\varepsilon}\|_{L^{2}}=1$. For the steady state problem
of (4.30), one can choose small $\tau>0$ and use $\tau\phi_{\varepsilon}$ as a
lower solution and $\frac{N}{|\Omega|}+\varepsilon$ as an upper solution to
establish the existence of a positive solution $I^{\varepsilon}$ in
$(0,\frac{N}{|\Omega|}+\varepsilon)$. The uniqueness and global asymptotic
stability can be proved similarly as in the argument of Lemma 4.3. Thus, we
have

$0\leq\limsup_{t\to\infty}I(\cdot,t)\leq\lim_{t\to\infty}\bar{I}(\cdot,t)=I^{\varepsilon}(\cdot),$
(4.43)

uniformly on $\overline{\Omega}$. On the other hand, notice that
$\\{I^{\varepsilon}\\}$ is uniformly bounded for all small $\varepsilon>0$. As
before, along a subsequence of $\varepsilon\to 0$, it holds
$I^{\varepsilon}\to I^{0}\geq 0$ in $C^{2}(\overline{\Omega})$. The fact
$\lambda_{*}=0$ enforces that the limiting function $I^{0}$ must be zero,
since otherwise one can easily derive a contradiction as in (4.42). Thus,
(4.41) follows from (4.43) and (4.29).

(ii) Since $\mathcal{R}_{0}>1$, we have $\lambda_{*}<0$ and
$\lambda_{\varepsilon}<0$ for all small $\varepsilon>0$. Then both (4.30) and
(4.31) admit a unique positive equilibrium which is globally asymptotically
stable. The rest of the argument is quite similar to that of Theorem 4.4 and
hence is omitted. ∎

######  Remark 4.7.

For model (4.20) with $0<p<1$, our findings in Theorems 4.2 and 4.4 strengthen
the results in [40, Theorem 2.5], where only uniform persistence property of
solutions is shown. For (4.20) with $p=1$, the global stability results in
Theorems 4.5 and 4.6 confirm the conjecture in [40, Remark 2.3 (i)] and thus
extends [13, Theorem 4.1].

## Acknowledgment

H. Li was partially supported by National Key R&D Program of China (No.
2021YFA1002100), NSF of China (No. 11971498), the project of Guangzhou
Science, Technology and Innovation Commission (No. 202102020807). T. Xiang was
partially supported by NSF of China (Nos. 12071476 and 11871226).

The authors would like to thank Prof. Rui Peng from Jiangsu Normal University,
China for many useful discussions.

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