linecolor=gray,linewidth=0.8pt figurem

# Uniform-in-time boundedness in a class of local and nonlocal nonlinear
attraction-repulsion chemotaxis models with logistics

Alessandro Columbu, Rafael Diaz Fuentes, Silvia Frassu⋆

###### Abstract.

The following fully nonlinear attraction-repulsion and zero-flux chemotaxis
model is studied:

($\Diamond$) $\begin{cases}u_{t}=\nabla\cdot\left((u+1)^{m_{1}-1}\nabla u-\chi
u(u+1)^{m_{2}-1}\nabla v+\xi u(u+1)^{m_{3}-1}\nabla w\right)+\lambda u-\mu
u^{r}&\textrm{ in }\Omega\times(0,T_{max}),\\\ \tau v_{t}=\Delta
v-\phi(t,v)+f(u)&\textrm{ in }\Omega\times(0,T_{max}),\\\ \tau w_{t}=\Delta
w-\psi(t,w)+g(u)&\textrm{ in }\Omega\times(0,T_{max}).\end{cases}$

Herein, $\Omega$ is a bounded and smooth domain of $\mathbb{R}^{n}$, for
$n\in\mathbb{N}$, $\chi,\xi,\lambda,\mu,r$ proper positive numbers,
$m_{1},m_{2},m_{3}\in\mathbb{R}$, and $f(u)$ and $g(u)$ regular functions that
generalise the prototypes $f(u)\simeq u^{k}$ and $g(u)\simeq u^{l}$, for some
$k,l>0$ and all $u\geq 0$. Moreover, $\tau\in\\{0,1\\}$, and
$T_{max}\in(0,\infty]$ is the maximal interval of existence of solutions to
the model. Once suitable initial data $u_{0}(x),\tau v_{0}(x),\tau w_{0}(x)$
are fixed, we are interested in deriving sufficient conditions implying
globality (i.e., $T_{max}=\infty$) and boundedness (i.e., $\lVert
u(\cdot,t)\rVert_{L^{\infty}(\Omega)}+\lVert
v(\cdot,t)\rVert_{L^{\infty}(\Omega)}+\lVert
w(\cdot,t)\rVert_{L^{\infty}(\Omega)}\leq C$ for all $t\in(0,\infty)$) of
solutions to problem ($\Diamond$ ‣ Uniform-in-time boundedness in a class of
local and nonlocal nonlinear attraction-repulsion chemotaxis models with
logistics). This is achieved in these scenarios:

  * $\triangleright$

For $\phi(t,v)$ proportional to $v$ and $\psi(t,w)$ to $w$, whenever $\tau=0$
and provided one of the following conditions

    1. (I)

$m_{2}+k<m_{3}+l$, $m_{2}+k<r$, $m_{2}+k<m_{1}+\frac{2}{n}$

is accomplished or $\tau=1$ in conjunction with one of these restrictions

    1. (i)

$\max{\\{m_{2}+k,m_{3}+l\\}}<r$,
$\max{\\{m_{2}+k,m_{3}+l\\}}<m_{1}+\frac{2}{n}$,

    2. (ii)

$m_{2}+k<r$ and $m_{3}+l<m_{1}+\frac{2}{n}$, $m_{2}+k<m_{1}+\frac{2}{n}$ and
$m_{3}+l<r$;

  * $\triangleright$

For $\phi(t,v)=\frac{1}{\left\lvert\Omega\right\rvert}\int_{\Omega}f(u)$ and
$\psi(t,w)=\frac{1}{\left\lvert\Omega\right\rvert}\int_{\Omega}g(u)$, whenever
$\tau=0$ if moreover one among (I), (I), (I) is fulfilled.

Our research improves and extends some results derived in [15, 26, 3, 6].

###### Key words and phrases:

Chemotaxis, Attraction-repulsion, Nonlinear production, Boundedness.
⋆Corresponding author<EMAIL_ADDRESS>
###### 2020 Mathematics Subject Classification:

Primary: 35K55, 35Q92. Secondary: 92C17.

Dipartimento di Matematica e Informatica

Università di Cagliari

Via Ospedale 72, 09124. Cagliari (Italy)

##  1\. Introduction and motivations

###  1.1. Some indications on attraction-repulsion chemotaxis models

The general formulation of an attraction-repulsion chemotaxis model,
incorporating logistic sources and linear or nonlinear productions, can be
expressed as follows:

(1) $\begin{cases}u_{t}=\nabla\cdot\left(D(u)\nabla u-S(u)\nabla v+T(u)\nabla
w\right)+h(u)&\text{ in }\Omega\times(0,T_{max}),\\\ \tau v_{t}=\Delta
v-\phi(t,v)+f(u)&\text{ in }\Omega\times(0,T_{max}),\\\ \tau w_{t}=\Delta
w-\psi(t,w)+g(u)&\textrm{ in }\Omega\times(0,T_{max}),\\\
u_{\nu}=v_{\nu}=w_{\nu}=0&\text{ on }\partial\Omega\times(0,T_{max}),\\\
u(x,0)=u_{0}(x),\;\tau v(x,0)=\tau v_{0}(x),\;\tau w(x,0)=\tau
w_{0}(x)&x\in\bar{\Omega},\\\ \end{cases}$

where $\Omega$ is a bounded and smooth domain in $\mathbb{R}^{n}$,
$n\in\mathbb{N}$, and $D(u)$, $S(u)$, $T(u)$, $h(u)$, $\phi(t,v)$,
$\psi(t,w)$, $f(u)$ and $g(u)$ are functions with specific regularity
properties. Furthermore, $\tau\in\\{0,1\\}$ and additional regular initial
data $u_{0}(x)\geq 0$ and $\tau v_{0}(x),\tau w_{0}(x)\geq 0$ are provided.
The subscript $\nu$ in $(\cdot)_{\nu}$ denotes the outward normal derivative
on $\partial\Omega$ and $T_{max}\in(0,\infty]$ represents the maximal temporal
instant up to which solutions to the system do exist.

This model holds practical significance since it is applicable to studying
inflammation in Alzheimer’s disease. In this frame, microglia secrete both
attractive and repulsive chemicals, and the system above describes the overall
mechanisms of the involved quantities. In the specific linear diffusion,
sensitivities and productions case (i.e., $D(u)=1$, $S(u)=\chi u$, $T(u)=\xi
u$, and $f(u)=g(u)=u$), and in the absence of dampening external terms (i.e.,
$h(u)\equiv 0$), for $\phi(t,v)=v$ and $\psi(t,w)=w$ in [20] the authors
provide insights connected to gathering mechanisms for (1) and develop
numerical analyses within bounded intervals, particularly when $\tau=0$.

###  1.2. The attractive and the repulsive models

Model (1) results from a combination of these perturbed signal-production
mechanisms with aggregative effect

(2) $u_{t}=\nabla\cdot(D(u)\nabla u-S(u)\nabla
v)+h(u)\quad\textrm{and}\quad\tau v_{t}=\Delta v-\phi(t,v)+f(u),\quad\textrm{
in }\Omega\times(0,T_{max}),$

and repulsive one

(3) $u_{t}=\nabla\cdot(D(u)\nabla u+T(u)\nabla
w)+h(u)\quad\textrm{and}\quad\tau w_{t}=\Delta w-\psi(t,w)+g(u),\quad\textrm{
in }\Omega\times(0,T_{max}).$

To properly understand model (1), it is essential to connect it with the
biological mechanisms described in problems (2) and (3). The related partial
differential equations are primarily employed to depict phenomena involving
the spatial-temporal distribution of unicellular organisms (denoted as
$u=u(x,t)$) within a confined and impenetrable environment ($\Omega$, with
$(\cdot)_{\nu}=0$ on $\partial\Omega$). The movement of these organisms is
influenced not only by natural diffusion (i.e., $\nabla\cdot D(u)\nabla u$),
but also by the gradient of a chemical signal $v=v(x,t)$, referred to as a
chemoattractant, which is produced at a rate $f(u)$ and leads to the
aggregation of cells through the cross-diffusion $-\nabla\cdot S(u)\nabla v$.
Additionally, a chemorepellent $w=w(x,t)$, secreted at a rate $g(u)$,
contributes to the cell repulsion by means of the counterpart $+\nabla\cdot
T(u)\nabla w$; moreover an external source $h(u)$ with both increasing and
decreasing effects on the cell distribution takes part in the phenomena.
Naturally, in order to have well-posed systems, some initial configurations
for cell density and chemical signals must be given; these are denoted by
$u(x,0)=u_{0}(x)$, $\tau v(x,0)=\tau v_{0}(x)$, and $\tau w(x,0)=\tau
w_{0}(x)$.

Regarding model (2), for the specific choice $D(u)=1$, $S(u)=\chi u$,
$\phi(t,v)=v$, $f(u)=u$ and $h(u)=0$ (for which essentially the chemical
signal $v$ increases with $u$), the natural homogenizing effect of the
diffusion might be insufficient to make that the cell density equally
disperses in the environment; indeed the drift term may force the system to
experience a gathering process, resulting in the formation of highly
concentrated spikes at some instants. This phenomenon, called chemotactic
collapse or blow-up at finite time, is intimately related to the value $\chi
m$ (being $m=\int_{\Omega}u_{0}$ the initial mass of the particle
distribution) and the dimension $n$. Mathematically, $T_{max}$ is finite and
the solution $(u,v)$ becomes unbounded at $T_{max}$. Notably, when $n=1$,
blow-up phenomena are excluded and in this case $T_{max}=\infty$ and the
solution $(u,v)$ is bounded. However, for $n\geq 2$, chemotactic collapse
occurs when $m\chi$ surpasses a critical value, herein denoted by $m_{\chi}$.
If $m\chi$ is lower than $m_{\chi}$, no instability appears in the cells’
motion. These findings are part of broader analyses, which explore existence
and properties (globality, uniform boundedness, or blow-up) of solutions to
the boundary-value problem associated to (2), especially in the parabolic-
elliptic version ($\tau=0$). Further details can be found in references such
as [25, 10, 14, 24, 36] and others.

If in problem (2) we fix $D(u)=1$, $S(u)=\chi u$, $\phi(t,v)=v$, $h(u)=0$, and
$\tau=1$, by replacing the linear segregation $f(u)=u$ with a nonlinear one of
the type $f(u)\simeq u^{k}$, with $0<k<\frac{2}{n}$ (for $n\geq 1$), implies
(see [18]) that all solutions remain bounded. On the contrary, when $\tau=0$
and $\phi(t,v)=\frac{1}{|\Omega|}\int_{\Omega}u$, it is known that in
spatially radial contexts (see [40]), $\frac{2}{n}$ plays the role of critical
value; indeed, solutions are bounded for any $n\geq 1$ and $0<k<\frac{2}{n}$,
while blow-up phenomena may occur when $k>\frac{2}{n}$. (As it will be
specified below, the term $\phi(t,v)=\frac{1}{|\Omega|}\int_{\Omega}u$
influences on the production of $v$ in a nonlocal way: see Remark 1.)

On the other hand, in the case of linear production rate, combining (2) with
terms representing population growth or decay, such as logistic sources (see
[31]), seems quite natural. The equation for the particle density becomes

$u_{t}=\Delta u-\chi\nabla\cdot(u\nabla v)+h(u),$

with $h(u)$ generally taking the form $h(u)=\lambda u-\mu u^{r}$, where
$\lambda,\mu>0$ and $r>1$. The mathematical intuition suggests that the
presence of a superlinear dampening effect may lead to boundedness and
smoothness. This has been established for large $\mu$ (especially when $r=2$,
$\phi(t,v)=v$, as shown in [30], [37] for respectively $\tau=0$ and $\tau=1$).
However, for certain values of $r>1$, blow-up has been demonstrated in the
parabolic-elliptic formulation, first with
$\phi(t,v)=\frac{1}{|\Omega|}\int_{\Omega}u$ for dimension $5$ or higher [38]
(see also [8] for a recent improvement), and later even in three-dimensional
domains, with $r<\frac{7}{6}$ and for the choice $\phi(t,v)=v$ (see [41]).

Finally, moving our attention to the situation where the produced signal $w$
has repulsive consequences on the cells’ motility, and this corresponds to
model (3), to the best of our knowledge no results regarding blow-up scenarios
are available. This is likely due to the repulsive nature of the mechanism,
and the existing literature on this topic is relatively limited, as seen in
references like [21] and [22] which analyse similar contexts.

###  1.3. An overview on the state of the art for the linear version of model
(1)

What we have discussed for the purely attractive mechanism (2) can be also
observed for attractive-repulsive models with linear diffusion and drift
terms. To be more precise, when in system (1) we set $D(u)=1$, $S(u)=\chi u$,
$T(u)=\xi u$, results dealing with blow-up and boundedness of solutions have
been derived for specific expressions of the equations for $v$ and $w$. We
mention two situations, known in the literature as local and nonlocal models.

### The local model: $\phi(t,v)=\beta v$, $\psi(t,w)=\delta w$

  * $\blacktriangleright$

The nonlogistic case ($h(u)=0$)

    * $\triangleright$

The case $\tau=0$: In the case of linear growth for both the chemoattractant
and the chemorepellent, i.e., $f(u)=\alpha u$, $\alpha>0$, and $g(u)=\gamma
u$, $\gamma>0$, the difference $\Theta:=\chi\alpha-\xi\gamma$ between the
parameters describing the impacts of the attraction and the repulsion, plays a
crucial role. Specifically, when $\Theta<0$ (indicating a regime where the
repulsion dominates the attraction), all solutions to the model are globally
bounded in any dimension. Conversely, when $\Theta>0$ (emphasizing now, that
the attraction is the dominating effect) and $n=2$, unbounded solutions can be
detected (for further details, see [9, 17, 28, 32, 43]). On the other hand,
for more general expressions of the production laws $f$ and $g$, and more
exactly those generalising the prototypes $f(u)\simeq\alpha u^{k}$, $k>0$, and
$g(u)\simeq\gamma u^{l}$, $l>0$, the following recent results valid for $n\geq
2$ apply ([33]): For any $\alpha,\beta,\gamma,\delta,\chi>0$, and $l>k\geq 1$
(or $k>l\geq 1$), there exists $\xi^{*}>0$ (or $\xi_{*}>0$) such that if
$\xi>\xi^{*}$ (or $\xi\geq\xi_{*}$), any sufficiently regular initial datum
$u_{0}(x)\geq 0$ (or $u_{0}(x)\geq 0$ small in some Lebesgue space) leads to a
unique classical solution which remains uniformly bounded in time. Moreover,
the same conclusion holds for any $\alpha,\beta,\gamma,\delta,\chi,\xi>0$, and
any sufficiently regular $u_{0}(x)\geq 0$ if the conditions $0<k<1$ and $l=1$
are satisfied (see also [6] for further improvements in some situations).

    * $\triangleright$

The case $\tau=1$: The research in [28] establishes that in two-dimensional
domains, when $f(u)=\alpha u$ and $g(u)=\gamma u$, sufficiently smooth initial
data lead to solutions that are globally bounded in time. This holds true if
the condition $\Theta<0$ is met (where $\Theta=\chi\alpha-\xi\gamma$) and
either

$\beta=\delta\quad\text{or}\quad-\frac{\chi^{2}\alpha^{2}(\beta-\delta)^{2}}{2\Theta\beta^{2}}\int_{\Omega}u_{0}(x)\leq
C(\Omega),\quad\textrm{for some constant }\;C(\Omega)>0.$

Additionally, in three-dimensional ball-shaped domains, [16] highlights the
occurrence of blow-up at a finite time under the condition
$\Theta=\chi\alpha-\xi\gamma>0$.

  * $\blacktriangleright$

The logistic case ($h(u)=\lambda u-\mu u^{r}$, $\lambda,\mu>0$, $r>1$)

    * $\triangleright$

The case $\tau=0$: For linear rates of the chemoattractant and the
chemorepellent, $f(u)=\alpha u$, $\alpha>0$, and $g(u)=\gamma u$, $\gamma>0$,
finite time blow-up is proved in [2] under the assumption
$\Theta=\chi\alpha-\xi\gamma>0$ in $n$-dimensional balls ($n\geq 3$), for some
$r=r(n)$ close to $1$ (additionally, also estimates of the blow-up time are
given). Otherwise, for nonlinear behaviour of $f$ and $g$ ($f(u)\simeq u^{k}$,
$g(u)\simeq u^{l}$), in [11] the authors show, inter alia, that if
$k<\max\\{l,r-1,\frac{2}{n}\\}$ all solutions are globally bounded; the
related long-time behaviour of these solutions is studied in [44].

    * $\triangleright$

The case $\tau=1$: For $r=2$ and in the three-dimensional setting whenever
$\mu$ is sufficiently large, boundedness and rate of convergence to constant
equilibria are discussed in [27].

### The nonlocal model: $\tau=0$,
$\phi(t,v)=\frac{1}{\lvert\Omega\rvert}\int_{\Omega}f(u)$,
$\psi(t,w)=\frac{1}{\lvert\Omega\rvert}\int_{\Omega}g(u)$

In [19] it is proved, together with other results, that for $f(u)\simeq\alpha
u^{k}$ and $g(u)\simeq\gamma u^{l}$ with $k>\frac{2}{n}$ and $k>l$, unbounded
solutions can be detected, in absence of logistics. By contrast, also in the
case $h\equiv 0$, in [6] the authors show the boundedness of solutions
provided that $k<l$ or $k=l$ and $\Theta:=\chi\alpha-\xi\gamma<0$, or
$l=k\in(0,\frac{2}{n})$ and $\Theta\geq 0$.

So far, we only restricted our discussion to linear diffusion and
sensitivities in model (1). Conversely, this project aims at providing results
for alike models (both including local and nonlocal effects) involving
nonlinear diffusion and drift terms, whose prototypes are $D(u)\simeq
u^{m_{1}},S(u)\simeq u^{m_{2}},T(u)\simeq u^{m_{3}}$, where
$m_{1},m_{2},m_{3}\in\mathbb{R}$.

##  2\. The nonlinear local and nonlocal models, known results and main claims

From the pure mathematical point of view, our attention is direct to these two
systems, one of local type (indicated with ($\mathcal{L}$)) and the other of
nonlocal nature, ($\mathcal{N}$):

($\mathcal{L}$) $\begin{dcases}u_{t}=\nabla\cdot\left((u+1)^{m_{1}-1}\nabla
u-\chi u(u+1)^{m_{2}-1}\nabla v+\xi u(u+1)^{m_{3}-1}\nabla w\right)+\lambda
u-\mu u^{r}&\text{in }\Omega\times(0,T_{max}),\\\ \tau v_{t}=\Delta v-\beta
v+f(u)&\text{in }\Omega\times(0,T_{max}),\\\ \tau w_{t}=\Delta w-\delta
w+g(u)&\text{in }\Omega\times(0,T_{max}),\\\
u_{\nu}=v_{\nu}=w_{\nu}=0&\text{on }\partial\Omega\times(0,T_{max}),\\\
u(x,0)=u_{0}(x),\tau v(x,0)=\tau v_{0}(x),\tau w(x,0)=\tau
w_{0}(x)&x\in\bar{\Omega},\end{dcases}$

and

($\mathcal{N}$) $\begin{dcases}u_{t}=\nabla\cdot\left((u+1)^{m_{1}-1}\nabla
u-\chi u(u+1)^{m_{2}-1}\nabla v+\xi u(u+1)^{m_{3}-1}\nabla w\right)+\lambda
u-\mu u^{r}&\text{in }\Omega\times(0,T_{max}),\\\ 0=\Delta
v-\frac{1}{\lvert\Omega\rvert}\int_{\Omega}f(u)+f(u)&\text{in
}\Omega\times(0,T_{max}),\\\ 0=\Delta
w-\frac{1}{\lvert\Omega\rvert}\int_{\Omega}g(u)+g(u)&\text{in
}\Omega\times(0,T_{max}),\\\ u_{\nu}=v_{\nu}=w_{\nu}=0&\text{on
}\partial\Omega\times(0,T_{max}),\\\ u(x,0)=u_{0}(x)&x\in\bar{\Omega},\\\
\int_{\Omega}v(x,t)\,dx=\int_{\Omega}w(x,t)\,dx=0&\text{for all
}t\in(0,T_{max}).\end{dcases}$

######  Remark 1.

In problem ($\mathcal{L}$), the equation $\tau v_{t}=\Delta v-\beta v+f(u)$
describes the local interaction between the involved quantities at each
spatial point $x$; contrarily, in ($\mathcal{N}$) the nonlocal term
$\frac{1}{|\Omega|}\int_{\Omega}f(u)$ appears, and it takes into account the
entire distribution of $u$ all over the domain $\Omega$. (Naturally, the same
observation can be done for the equations related to $w$.)

Additionally, let us clarify that in the nonlocal model, $v$ and $w$ stand for
the deviations of the chemoattractant and the chemorepellent (and not for the
chemicals themselves). Since by definition the deviation is the difference
between the signal concentration and its mean value, it follows that the means
of $v$ and $w$ vanish, exactly as specified in the last positions of problem
($\mathcal{N}$). As a consequence, $v$ and $w$ change sign.

Consistently with the above problems, this paper is contextualized in the
frame of a series of results dealing with local and nonlocal, and linear and
nonlinear, attraction-repulsion chemotaxis systems; in particular this
research improves and extends already known analyses in the literature (we
will give more details in the sequel).

In the specific, as far as we know, for such discussed fully nonlinear
versions, the literature is rather poor; nevertheless, issues dealing with
boundedness and blow-up of solutions have been addressed. More precisely, for
problem ($\mathcal{L}$) we mention the works [26] and [3], where in the first
the authors ensure the boundedness of solutions, for both $\tau=0$ and
$\tau=1$, under suitable assumptions on the data; in the second boundedness
and blow-up analyses, in this case for the simplified parabolic-elliptic-
elliptic version, are discussed. For the nonlocal problem ($\mathcal{N}$), and
in the context of nonlinear productions, in [34] blow-up scenarios have been
detected for $m_{1}\in\mathbb{R}$ and $m_{2}=m_{3}=1$; in turn, in the recent
paper [5] this result has been extended to the case $m_{1}\in\mathbb{R}$ and
$m_{2}=m_{3}\in\mathbb{R}$. As to the boundedness counterpart, at this point
we simply make reference to [15]. (Since in this research we will improve some
results derived in [15, 26, 3, 6], we will spend more words on these papers in
the below Remark 2 and Remark 3.)

###  2.1. Presentation of the main theorems

In order to present our claims, some positions have to be previously fixed. In
particular, here we assume that $\Omega$, sources $f,g$ and the initial data
$u_{0}=u_{0}(x),\tau v_{0}=\tau v_{0}(x)$ and $\tau w_{0}=\tau w_{0}(x)$ are
such that

(4) $\begin{cases}\Omega\subset\mathbb{R}^{n},n\in\mathbb{N},\textrm{ is a
bounded domain with smooth boundary $\partial\Omega$},\\\
f,g:[0,\infty)\rightarrow\mathbb{R}^{+},\textrm{ with }f,g\in
C^{1}([0,\infty)),\\\ u_{0},\tau v_{0},\tau
w_{0}:\bar{\Omega}\rightarrow\mathbb{R}^{+},\textrm{ with }u_{0},\tau
v_{0},\tau w_{0}\in W^{1,\infty}(\Omega);\end{cases}$

moreover, we suppose, for $\alpha,l,k>0$ and $0<\gamma_{0}\leq\gamma_{1}$,

(5) $0\leq f(s)\leq\alpha(s+1)^{k}\quad\text{and}\quad\gamma_{0}(s+1)^{l}\leq
g(s)\leq\gamma_{1}(s+1)^{l}.$

Additionally, we will frequently invoke these

######  Assumptions 2.1.

For $n\in\mathbb{N}$, $m_{1},m_{2},m_{3}\in\mathbb{R}$, $k,l>0$ and $r>1$ let
us set:

  1. ($\mathcal{A}_{1}$)

$m_{2}+k<m_{3}+l$; $m_{2}+k<r$; $m_{2}+k<m_{1}+\frac{2}{n}$; $m_{3}+l<r$;
$m_{3}+l<m_{1}+\frac{2}{n}$.

The above preparations allow us to give the following two theorems.

######  Theorem 2.2.

For $\tau=0$, let $\Omega$, $f,g,u_{0}$ comply with hypotheses in (4), (5),
$\lambda,\mu,\chi,\xi,\beta,\delta>0$ and $q>n$. Additionally, let one among
($\mathcal{A}_{1}$), ($\mathcal{A}_{1}$), ($\mathcal{A}_{1}$) in Assumptions
2.1 hold true. Then problems ($\mathcal{L}$) and ($\mathcal{N}$) admit a
unique solution

$u\in C^{0}(\bar{\Omega}\times[0,\infty))\cap
C^{2,1}(\bar{\Omega}\times(0,\infty))\;\textrm{and}\;v,w\in
C^{2,0}(\bar{\Omega}\times(0,\infty))\cap
L^{\infty}_{loc}((0,\infty);W^{1,q}(\Omega)),$

such that $u$ is nonnegative and $u,v,w$ are bounded on
$\bar{\Omega}\times[0,\infty)$. In particular, $v,w$ are as well nonnegative
on $\bar{\Omega}\times[0,\infty)$ for problem ($\mathcal{L}$).

######  Theorem 2.3.

For $\tau=1$, let the remaining hypotheses of Theorem 2.2 be satisfied.
Additionally, let one among ($\mathcal{A}_{1}$) and ($\mathcal{A}_{1}$)
jointly with one between ($\mathcal{A}_{1}$) and ($\mathcal{A}_{1}$) in
Assumptions 2.1 hold true. Then problem ($\mathcal{L}$) admits a unique
solution

$u\in C^{0}(\bar{\Omega}\times[0,\infty))\cap
C^{2,1}(\bar{\Omega}\times(0,\infty))\;\textrm{and}\;v,w\in
C^{0}(\bar{\Omega}\times[0,\infty))\cap
C^{2,1}(\bar{\Omega}\times(0,\infty))\cap
L^{\infty}_{loc}((0,\infty);W^{1,q}(\Omega)),$

such that $u,v,w$ are nonnegative and bounded on
$\bar{\Omega}\times[0,\infty)$.

As anticipated in the introductory part, let us specify in which sense our
results improve what known so far in the literature.

######  Remark 2.

(Comparisons with nonlocal problem ($\mathcal{N}$)) Let us make these
observations.

  * $\blacktriangleright$

Nonlinear diffusion and sensitivities ($m_{1},m_{2},m_{3}\in\mathbb{R}$): in
[15, Theorem 1.1] boundedness for problem ($\mathcal{N}$) is achieved for each
of the following cases:

    1. i)

$m_{3}+l<m_{2}+k+1$ and $m_{2}+k<m_{1}+\frac{2}{n}$;
$m_{2}+k<m_{3}+l<m_{1}+\frac{2}{n}$;
$\max\\{m_{2}+k,m_{3}+l\\}<r<m_{1}+\frac{2}{n}$.

It is seen that assumption ($\mathcal{A}_{1}$) in Theorem 2.2 is sharper with
respect to i), and ($\mathcal{A}_{1}$) with respect to i) and, finally,
($\mathcal{A}_{1}$) with respect to i).

  * $\blacktriangleright$

Linear diffusion and sensitivities ($m_{1}=m_{2}=m_{3}=1$): in [6, Theorem
2.3] boundedness for problem ($\mathcal{N}$) is established, among other
situations, whenever:

    1. (a)

$k<l$; $k=l\in(0,\frac{2}{n})$ and $\Theta_{0}:=\chi\alpha-\xi\gamma_{0}\geq
0$.

Evidently, if from the one hand (a) is recovered from ($\mathcal{A}_{1}$), on
the other hand, also in this case some milder conclusions can be observed: in
the specific ($\mathcal{A}_{1}$) improves (a) and provides a further situation
where $l>0$ and $k\in\left(0,\frac{2}{n}\right)$.

######  Remark 3.

(Comparisons with local problem ($\mathcal{L}$)) Even for the local problem,
the present analysis gives some more precise insight in the context of
attraction-repulsion chemotaxis mechanisms.

  * $\blacktriangleright$

Nonlinear diffusion and sensitivities ($m_{1},m_{2},m_{3}\in\mathbb{R}$):

    * $\triangleright$

$\tau=0$: In [3, Theorem 3.1 and Theorem 3.5], where the authors consider the
problem with linear productions ($k=l=1$), globality of solutions is carried
out for the case $m_{2}\leq m_{3}$, so leaving room to the case $m_{2}>m_{3}$;
in particular, for either $m_{2}<m_{3}$ or $m_{2}=m_{3}$ and
$\Theta_{0}=\chi\alpha-\xi\gamma_{0}<0$ (naturally, for $k=l=1$ we have
$\gamma_{0}=\gamma_{1}$), boundedness is, respectively, achieved. Indeed,
assumptions ($\mathcal{A}_{1}$) and ($\mathcal{A}_{1}$) of Theorem 2.2 cover
the case $m_{2}>m_{3}$.

Furthermore, Theorem 2.2 extends [26, Theorem 1.3], where only the case
$m_{1}\in\mathbb{R}$, $m_{2}=m_{3}=1$ and $k,l\geq 1$ is addressed;
particularly assumption ($\mathcal{A}_{1}$) is less restricted than [26,
Theorem 1.3, (i)].

    * $\triangleright$

$\tau=1$: In [26, Theorem 1.1] boundedness of solutions is established, inter
alia, if

(6)
$\max\\{m_{2}+k,m_{3}+l\\}<m_{1}+\frac{2}{n}\quad\text{or}\quad\max\\{m_{2}+k,m_{3}+l\\}<r$

are fulfilled. In this way, (($\mathcal{A}_{1}$), ($\mathcal{A}_{1}$)) and
(($\mathcal{A}_{1}$), ($\mathcal{A}_{1}$)) of Theorem 2.3 coincide with (6),
but (($\mathcal{A}_{1}$), ($\mathcal{A}_{1}$)) and (($\mathcal{A}_{1}$),
($\mathcal{A}_{1}$)) provide a further scenario towards boundedness.

  * $\blacktriangleright$

Linear diffusion and sensitivities ($m_{1}=m_{2}=m_{3}=1$): For $\tau=0$ if we
compare Theorem 2.2 with [6, Theorem 2.1], not only the same conclusion given
in the second item of Remark 2 still applies but even [6, Thorem 2.1, (2)] is
turned into a weaker condition. For the fully parabolic case, assumption
($\mathcal{A}_{1}$) and ($\mathcal{A}_{1}$) of Theorem 2.3 imply
$l,k\in(0,\frac{2}{n})$, so improving [6, Theorem 2.2].

##  3\. Adjusting parameters and recalling necessary results

Let us now dedicate to summarise some tools which will be used in our
reasoning. These are connected to algebraic inequality and regularity results
for Partial Differential Equations.

######  Lemma 3.1.

Let $A,B\geq 0$ and $d_{1},d_{2}>0$ be such that $d_{1}+d_{2}<1$. Then for all
$\epsilon>0$ there exists $c=c_{\varepsilon}>0$ such that

$A^{d_{1}}B^{d_{2}}\leq\epsilon(A+B)+c.$

###### Proof.

The proof can be found in [7, Lemma 4.3]. ∎

######  Lemma 3.2.

Let $\Omega\subset\mathbb{R}^{n}$ satisfy condition in (4), $\eta>0$ and
$q>\max\\{n,\frac{1}{\eta}\\}$.

  * $\diamond$

_Elliptic regularity_ : Let $\vartheta\in(0,1)$. If $\psi\in
C^{\vartheta}(\bar{\Omega})$, then the solution $z\in
C^{2+\vartheta}(\bar{\Omega})$ of

$\begin{cases}-\Delta z+\eta z=\psi&\text{in }\Omega,\\\ z_{\nu}=0&\text{on
}\partial\Omega,\end{cases}$

is such that

(7) $\nabla z\in L^{\infty}(\Omega).$

Moreover for any $\hat{c},\rho>0$, there is $C_{\rho}>0$ such that

(8)
$\hat{c}\int_{\Omega}z^{q}\leq\rho\int_{\Omega}\psi^{q}+C_{\rho}\left(\int_{\Omega}\psi\right)^{q}.$

  * $\diamond$

_Parabolic regularity_ : Let $T\in(0,\infty]$. Then, for $\psi\in
L^{q}([0,T);L^{q}(\Omega))$ and $z_{0}\in W^{2,q}(\Omega)$ with
$\partial_{\nu}z_{0}=0$ on $\partial\Omega$, there exists
$C_{P}=C_{P}(\Omega,q,\rVert z_{0}\rVert_{W^{2,q}(\Omega)})$ such that every
solution $z\in W^{1,q}_{loc}([0,T);L^{q}(\Omega))\cap
L^{q}_{loc}([0,T);W^{2,q}(\Omega))$ of

$\begin{cases}z_{t}=\Delta z-\eta z+\psi&\text{in }\Omega\times(0,T),\\\
\partial_{\nu}z=0&\text{on }\partial\Omega\times(0,T),\\\
z(\cdot,0)=z_{0}&\text{on }\Omega,\end{cases}$

satisfies

(9) $\int_{0}^{t}e^{s}\int_{\Omega}\left\lvert\Delta
z(\cdot,s)\right\rvert^{q}\,ds\leq
C_{P}\left[1+\int_{0}^{t}e^{s}\int_{\Omega}\lvert\psi(\cdot,s)\rvert^{q}\,ds\right]\quad\text{for
all }t\in(0,T).$

Additionally, if $\psi\in L^{\infty}([0,T);L^{q}(\Omega))$ then

(10) $\nabla z\in L^{\infty}((0,T);L^{\infty}(\Omega)).$

###### Proof.

Relation (7) is an obvious consequence of [1, Theorem IX.33]. For estimate
(8), we indicate [39, Lemma 2.2] (and also [33, Lemma 3.1]) and we precise
that the power $q$ may be replaced by any other one larger than $1$. (Since
$q$ will be chosen arbitrarily large we preferred to synthesize the
nomenclature.) On the other hand, for (9) we refer to [13, Lemma 3.6], whereas
for (10) we can invoke [12, Lemma 4.1] and the embedding
$W^{1,\frac{nq}{(n-q)_{+}}}(\Omega)\subset L^{\infty}(\Omega)$, valid for
$q>n$. (The reason why we need $q>\frac{1}{\eta}$ is understandable seeing
[13, Lemma 3.6].) ∎

The following lemma defines crucial parameters; its proof is based on some of
the relations fixed in Assumptions 2.1.

######  Lemma 3.3.

Let $n,k,l,m_{1},m_{2},m_{3}$ be as in Assumptions 2.1 and let relations
($\mathcal{A}_{1}$) and ($\mathcal{A}_{1}$) be valid. Then, there exists
$\bar{p}>1$ such that, for all $p>\bar{p}$, $q>1$ and

$\theta(p)\coloneqq\frac{\frac{p+m_{1}-1}{2}-\frac{p+m_{1}-1}{2(p+m_{2}+k-1)}}{\frac{p+m_{1}-1}{2}-\frac{1}{2}+\frac{1}{n}},\quad\sigma(p)\coloneqq\frac{2(p+m_{2}+k-1)}{p+m_{1}-1},\quad\theta_{1}(p)\coloneqq\frac{\frac{p+m_{1}-1}{2}-\frac{p+m_{1}-1}{2(p+m_{3}+l-1)}}{\frac{p+m_{1}-1}{2}-\frac{1}{2}+\frac{1}{n}},\quad\sigma_{1}(p)\coloneqq\frac{2(p+m_{3}+l-1)}{p+m_{1}-1}$
$\theta_{2}(p)\coloneqq\frac{\frac{p+m_{1}-1}{2}-\frac{p+m_{1}-1}{2l}}{\frac{p+m_{1}-1}{2}-\frac{1}{2}+\frac{1}{n}},\quad\quad\theta_{3}(p)\coloneqq\frac{\frac{p+m_{1}-1}{2}-\frac{p+m_{1}-1}{2p}}{\frac{p+m_{1}-1}{2}-\frac{1}{2}+\frac{1}{n}},\quad\quad\theta_{4}(p)\coloneqq\frac{\frac{p+m_{1}-1}{2}-\frac{p+m_{1}-1}{2q}}{\frac{p+m_{1}-1}{2}-\frac{1}{2}+\frac{1}{n}},\quad\quad\sigma_{2}(p)\coloneqq\frac{2(p+q)}{p+m_{1}-1},$

these relations hold

(11a) $0<\theta<1,$

(11b) $0<\frac{\sigma\theta}{2}<1,$

(11c) $0<\theta_{1}<1,$

(11d) $0<\frac{\sigma_{1}\theta_{1}}{2}<1,$

(11e) $0<\theta_{2}<1,$

(11f) $0<\frac{\sigma_{1}\theta_{2}}{2}<1,$

(11g) $0<\theta_{4}<1,$

(11h) $0<\frac{\sigma_{2}\theta_{4}}{2}<1,$

(11i) $0<\theta_{3}<1,$

where (11e) and (11f) are valid only for $l>1$.

###### Proof.

First let us consider the function $\theta(p)$. Since
$\lim_{p\to\infty}\theta(p)=1,$ we have that $\theta(p)$ is definitively
positive; on the other hand, $\theta(p)$ increases for $p$ sufficiently large
so that 1 is an upper bound. Therefore, (11a) is proved. Relations (11c),
(11i), (11b) and (11d) follow by means of analogous arguments, once for (11b)
and (11d) conditions ($\mathcal{A}_{1}$) and ($\mathcal{A}_{1}$) are
respectively taken into account. Let us consider now the functions
$\theta_{2}(p)$ and $\sigma_{1}(p)$. Relations (11e) and (11f) are consequence
of

$\lim_{p\to\infty}\theta_{2}(p)=\lim_{p\to\infty}\frac{\sigma_{1}(p)\theta_{2}(p)}{2}=1-\frac{1}{l}\in(0,1),\quad\text{for
$l>1$}.$

In the same way, relations (11g) and (11h) are given for every $q>1$.

From all of the above, it is possible to find $\bar{p}>1$ such that the
previous relations are complied for every $p>\bar{p}$. ∎

##  4\. Local existence and extensibility criterion

A first necessary step on which our computations have to rely is the local-in-
time existence of classical solutions to systems ($\mathcal{L}$) and
($\mathcal{N}$). The succeeding requirement is, indeed, providing a criterion
capable to turn such solutions into global ones. The following lemma focuses
on these two aspects.

######  Lemma 4.1.

For $\tau\in\\{0,1\\}$, let $\Omega$, $f,g,u_{0},\tau v_{0}$ and $\tau w_{0}$
comply with hypotheses in (4). Moreover, let
$\chi,\xi,\beta,\delta,\lambda,\mu>0$, $m_{1},m_{2},m_{3}\in\mathbb{R}$,
$r>1$, and $q>n$. Then there exist $T_{max}\in(0,\infty]$ and a unique
solution $(u,v,w)$ to problems ($\mathcal{L}$) and ($\mathcal{N}$), defined in
$\Omega\times(0,T_{max})$ and such that if $\tau=0$

$u\in C^{0}(\bar{\Omega}\times[0,T_{max}))\cap
C^{2,1}(\bar{\Omega}\times(0,T_{max}))\;\textrm{and}\;v,w\in
C^{2,0}(\bar{\Omega}\times(0,T_{max}))\cap
L^{\infty}_{loc}((0,T_{max});W^{1,q}(\Omega)),$

whereas if $\tau=1$

$u\in C^{0}(\bar{\Omega}\times[0,T_{max}))\cap
C^{2,1}(\bar{\Omega}\times(0,T_{max}))\;\textrm{and}\;v,w\in
C^{0}(\bar{\Omega}\times[0,T_{max}))\cap
C^{2,1}(\bar{\Omega}\times(0,T_{max}))\cap
L^{\infty}_{loc}((0,T_{max});W^{1,q}(\Omega)).$

The components $(u,v,w)$ of solutions to problem ($\mathcal{L}$) are
nonnegative, whereas for ($\mathcal{N}$) only $u$ is nonnegative.

In addition,

(12) $\text{if}\quad T_{max}<\infty\quad\text{then}\quad\limsup_{t\to
T_{max}}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}=\infty.$

Finally, let $u\in L^{\infty}((0,T_{max}),L^{p}(\Omega))$ for all $p>1$; then
$u\in L^{\infty}((0,T_{max}),L^{\infty}(\Omega))$ and $T_{max}=\infty$.

###### Proof.

Existence and uniqueness of solutions can be established using well known
techniques based on fixed point arguments and elliptic and parabolic
regularity results: we refer to [4, 23, 35, 42].

Let us spend some words on the last implication. Naturally $u=u(x,t)$, with
$(x,t)\in\Omega\times(0,T_{max})$, also classically solves problem (A.1) in
[29, Appendix A] for

$D(x,t,u)=(u+1)^{m_{1}-1},\quad\tilde{f}(x,t)=-\chi u(u+1)^{m_{2}-1}\nabla
v+\xi u(u+1)^{m_{3}-1}\nabla w,\quad g(x,t)=\lambda u-\mu u^{r}.$

Specifically, by making use of the Neumann boundary conditions, we can see
that (A.2)–(A.5) are complied. On the other hand, for any $\lambda,\mu>0$ and
$r>1$, it holds that $\lambda u-\mu u^{r}$ has a positive maximum $L$ at
$u_{M}=\left(\frac{\lambda}{r\mu}\right)^{\frac{1}{r-1}}$, so that from
$g(x,t)\leq L$ in $\Omega\times(0,T_{max})$ the second inclusion (A.6) is
obtained for any $q_{2}>1$. Since, by hypotheses, $u\in
L^{\infty}((0,T_{max});L^{p}(\Omega))$ for every $p>1$, we have that
$\int_{\Omega}u^{p}$ is uniformly bounded on $(0,T_{max})$ for $p$ arbitrary
large (without relabelling it) and henceforth conditions (A.7)–(A.10) are
fulfilled. As to the first inclusion of [29, (A.6)], if $\tau=0$ in problem
($\mathcal{L}$) (or model ($\mathcal{N}$)), from the gained inclusion $u\in
L^{\infty}((0,T_{max});L^{p}(\Omega))$ we have $f(u)\in
L^{\infty}((0,T_{max});L^{p}(\Omega))$, and in turn relation (7) provides
$\nabla v\in L^{\infty}((0,T_{max});L^{\infty}(\Omega))$, and similarly
$\nabla w\in L^{\infty}((0,T_{max});L^{\infty}(\Omega))$. For $\tau=1$ we
directly invoke (10); in both cases we have that, for any $q_{1}>1$,
$\tilde{f}\in L^{\infty}((0,T_{max});L^{q_{1}}(\Omega))$. As a consequence of
what explained, we have the claim by virtue of [29, Lemma A.1] and the
extensibilty criterion (12). ∎

##  5\. Some a priori estimates

In this section we will dedicate to derive some uniform-in-time estimates of
the previously gained local solution the investigated problems. In this frame,
from now on we will tacitly assume that

  * $\triangleright$

all the constants $c_{i}$ ($i=1,2,\ldots$) appearing below are positive,

  * $\triangleright$

the triplet $(u,v,w)$ indicates the local solution to models ($\mathcal{L}$)
or ($\mathcal{N}$) (naturally recognizable from the context), obtained in
Lemma 4.1.

The forthcoming lemmas hold for models ($\mathcal{L}$) and ($\mathcal{N}$).

######  Lemma 5.1.

The component $u$ is such that the associated mass $\int_{\Omega}u(x,t)dx$ is
uniformly bounded over $(0,T_{max})$; more specifically,

$\int_{\Omega}u\leq
M:=\max\left\\{\displaystyle\int_{\Omega}u_{0}(x)\,dx,\left(\frac{\lambda}{\mu}|\Omega|^{r-1}\right)^{\frac{1}{r-1}}\right\\}\quad\text{for
all }t\in[0,T_{max}).$

###### Proof.

This property can be proved by integrating over $\Omega$ the first equation of
the models and then by applying the Hölder inequality and ODI comparison
principles. ∎

######  Lemma 5.2.

Let hypotheses of Lemma 3.3 be valid and $\bar{p}>1$ the value therein found.
Then for every $\hat{c},\varepsilon_{1},\varepsilon_{2}>0$ and for every
$p>\bar{p}$, there exist some constants
$c\textsubscript{1},c\textsubscript{2},c\textsubscript{3},c\textsubscript{4},c\textsubscript{5},c\textsubscript{6}$
such that

(13a)
$\hat{c}\int_{\Omega}(u+1)^{p+m_{2}+k-1}\leq\frac{p-1}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{1}\quad\text{on
}(0,T_{max}),\text{ provided \ref{M3}},$ (13b)
$\hat{c}\int_{\Omega}(u+1)^{p+m_{3}+l-1}\leq\frac{p-1}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{2}\quad\text{on
}(0,T_{max}),\text{ provided \ref{M5}},$ (13c)
$\hat{c}\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{l}\leq\varepsilon_{1}\int_{\Omega}(u+1)^{p+m_{3}+l-1}+\frac{p-1}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{3}\quad\text{on
}(0,T_{max}),$ (13d)
$\hat{c}\int_{\Omega}(u+1)^{p+m_{3}-1}w\leq\varepsilon_{2}\int_{\Omega}(u+1)^{p+m_{3}+l-1}+\frac{p-1}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{4}\quad\text{on
}(0,T_{max}),$ (13e)
$-\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\leq
c\textsubscript{5}-c\textsubscript{6}\left(\int_{\Omega}(u+1)^{p}\right)^{\frac{p+m_{1}-1}{p\theta_{1}}}\quad\text{on
}(0,T_{max}).$

###### Proof.

Let us show (13a). Under assumption ($\mathcal{A}_{1}$), by taking into
account (11a), (11b) and the boundedness of the mass (Lemma 5.1), we can
derive through the Gagliardo–Nirenberg and Young’s inequalities this bound

$\begin{split}\hat{c}\int_{\Omega}(u+1)^{p+m_{2}+k-1}=&\hat{c}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2(p+m_{2}+k-1)}{p+m_{1}-1}}(\Omega)}^{\frac{2(p+m_{2}+k-1)}{p+m_{1}-1}}\\\
\leq&c\textsubscript{7}\left(\left\lVert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{2}(\Omega)}^{\theta}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}^{1-\theta}+\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}\right)^{\sigma}\\\
\leq&c\textsubscript{8}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\right)^{\frac{\sigma\theta}{2}}+c\textsubscript{9}\leq\frac{p-1}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{1}\quad\text{on
}(0,T_{max}).\end{split}$

(We precise that we have made use of

(14) $(A+B)^{s}\leq 2^{s}(A^{s}+B^{s})\quad\text{for all }A,B,s>0,$

and we might employ this algebraic relation without mentioning it if not
necessary.)

Similarly to what we have done before, by supposing ($\mathcal{A}_{1}$),
estimate (13b) is obtained by applying in this case (11c) and (11d), so
entailing

$\begin{split}\hat{c}\int_{\Omega}(u+1)^{p+m_{3}+l-1}=&\hat{c}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2(p+m_{3}+l-1)}{p+m_{1}-1}}(\Omega)}^{\frac{2(p+m_{3}+l-1)}{p+m_{1}-1}}\\\
\leq&c\textsubscript{10}\left(\left\lVert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{2}(\Omega)}^{\theta_{1}}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}^{1-\theta_{1}}+\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}\right)^{\sigma_{1}}\\\
\leq&c\textsubscript{11}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\right)^{\frac{\sigma_{1}\theta_{1}}{2}}+c\textsubscript{12}\leq\frac{p-1}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{2}\quad\text{for
all }t\in(0,T_{max}).\end{split}$

In order to prove (13c) we distinguish the cases $l\leq 1$ and $l>1$. If
$0<l\leq 1$ the boundedness of the mass, given by Lemma 5.1, and Young’s
inequality yield

$\begin{split}\hat{c}\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{l}&\leq
c\textsubscript{13}\int_{\Omega}(u+1)^{p+m_{3}-1}\leq\varepsilon_{1}\int_{\Omega}(u+1)^{p+m_{3}+l-1}+c\textsubscript{3}\quad\text{on
}(0,T_{max}).\end{split}$

On the other hand, if $l>1$ we first have to introduce
$q>\max{\\{l,m_{3}+l-1\\}}$. Subsequently, by applying twice the Hölder
inequality, we get by relying on Lemma 3.1

(15)
$\begin{split}\hat{c}\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{l}&\leq
c\textsubscript{14}\left[\int_{\Omega}(u+1)^{p+m_{3}+l-1}\right]^{\frac{p+m_{3}-1}{p+m_{3}+l-1}}\left[\left(\int_{\Omega}(u+1)^{q}\right)^{\frac{p}{q}+1}\right]^{\frac{l}{p+q}}\\\
&\leq\varepsilon_{1}\int_{\Omega}(u+1)^{p+m_{3}+l-1}+\varepsilon_{1}\left(\int_{\Omega}(u+1)^{q}\right)^{\frac{p}{q}+1}+c\textsubscript{15}\quad\text{for
all }t\in(0,T_{max}).\end{split}$

Now we focus on the second integral of the right-hand side of (15). By
exploiting (11g), (11h) and Lemma 5.1, a combination of the
Gagliardo–Nirenberg and Young’s inequalities gives

$\begin{split}\varepsilon_{1}\left(\int_{\Omega}(u+1)^{q}\right)^{\frac{p}{q}+1}=&\varepsilon_{1}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2q}{p+m_{1}-1}}(\Omega)}^{\frac{2(p+q)}{p+m_{1}-1}}\\\
\leq&c\textsubscript{16}\left(\left\lVert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{2}(\Omega)}^{\theta_{4}}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}^{1-\theta_{4}}+\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}\right)^{\sigma_{2}}\\\
\leq&c\textsubscript{17}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\right)^{\frac{\sigma_{2}\theta_{4}}{2}}+c\textsubscript{18}\leq\frac{p-1}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{3}\quad\text{on
}(0,T_{max}),\end{split}$

which plugged in the previous one concludes the proof.

As to (13d), again by means of the Young inequality and relation (8),
recalling the definition of $g$ in (5) we can write

(16)
$\begin{split}\hat{c}\int_{\Omega}(u+1)^{p+m_{3}-1}w&\leq\rho\int_{\Omega}(u+1)^{p+m_{3}+l-1}+c\textsubscript{19}\int_{\Omega}w^{\frac{p+m_{3}-1}{l}+1}\\\
&\leq\rho\int_{\Omega}(u+1)^{p+m_{3}+l-1}+\rho\int_{\Omega}g(u)^{\frac{p+m_{3}+l-1}{l}}+C_{\rho}\left(\int_{\Omega}g(u)\right)^{\frac{p+m_{3}+l-1}{l}}\\\
&\leq\varepsilon_{2}\int_{\Omega}(u+1)^{p+m_{3}+l-1}+c\textsubscript{20}\left(\int_{\Omega}(u+1)^{l}\right)^{\frac{p+m_{3}+l-1}{l}}\quad\text{for
all }t\in(0,T_{max}).\end{split}$

For $l\leq 1$, due to Lemma 5.1 the last integral of the right-hand side of
(16) is bounded, and we have

$\hat{c}\int_{\Omega}(u+1)^{p+m_{3}-1}w\leq\varepsilon_{2}\int_{\Omega}(u+1)^{p+m_{3}+l-1}+c\textsubscript{4}\quad\text{on
}(0,T_{max}).$

Instead, for $l>1$, by virtue of (11e), Lemma 5.1 and (11f), the
Gagliardo–Nirenberg and Young’s inequalities imply

$\begin{split}c\textsubscript{20}\left(\int_{\Omega}(u+1)^{l}\right)^{\frac{p+m_{3}+l-1}{l}}&=c\textsubscript{20}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\sigma_{1}}_{L^{\frac{2l}{p+m_{1}-1}}(\Omega)}\\\
&\leq
c\textsubscript{21}\left(\left\lVert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\theta_{2}}_{L^{2}(\Omega)}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{1-\theta_{2}}_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}+\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}\right)^{\sigma_{1}}\\\
&\leq
c\textsubscript{22}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\right)^{\frac{\sigma_{1}\theta_{2}}{2}}+c\textsubscript{23}\\\
&\leq\frac{p-1}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{4}\quad\text{for
all }t\in(0,T_{max}),\end{split}$

and we conclude by inserting this gained bound into (16).

Finally, let us prove (13e). By taking into account (11i) and Lemma 5.1, a
further application of the Gagliardo–Nirenberg inequality leads to

$\begin{split}\int_{\Omega}(u+1)^{p}=&\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2p}{p+m_{1}-1}}(\Omega)}^{\frac{2p}{p+m_{1}-1}}\\\
\leq&c\textsubscript{24}\left(\left\lVert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{2}(\Omega)}^{\theta_{3}}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}^{1-\theta_{3}}+\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}\right)^{\frac{2p}{p+m_{1}-1}}\\\
\leq&c\textsubscript{25}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\right)^{\frac{p\theta_{3}}{p+m_{1}-1}}+c\textsubscript{26}\leq
c\textsubscript{27}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+1\right)^{\frac{p\theta_{3}}{p+m_{1}-1}}\quad\text{on
}(0,T_{max}),\end{split}$

obtaining the claim after basic manipulations in the previous estimate. ∎

######  Lemma 5.3.

For the value of $\bar{p}>1$ found in Lemma 3.3, let for $p>\bar{p}$ define
the functional $\varphi(t)$ by

$\varphi(t):=\frac{1}{p}\int_{\Omega}(u+1)^{p},$

and for $j\in\\{m_{2},m_{3}\\}$, let

(17) $F_{j}(u):=\int_{0}^{u}\hat{u}\left(\hat{u}+1\right)^{p+j-3}d\hat{u}.$

Then for all $t\in(0,T_{max})$ and some $c\textsubscript{28}$ and
$c\textsubscript{29}$

(18)
$\varphi^{\prime}(t)\leq-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}-(p-1)\chi\int_{\Omega}F_{m_{2}}(u)\Delta
v+(p-1)\xi\int_{\Omega}F_{m_{3}}(u)\Delta
w-c\textsubscript{28}\int_{\Omega}(u+1)^{p+r-1}+c\textsubscript{29}.$

###### Proof.

Let us differentiate the energy functional; we have

$\displaystyle\varphi^{\prime}(t)=$
$\displaystyle\int_{\Omega}(u+1)^{p-1}u_{t}$ $\displaystyle=$
$\displaystyle\int_{\Omega}(u+1)^{p-1}\nabla\cdot\left((u+1)^{m_{1}-1}\nabla
u\right)-\chi\int_{\Omega}(u+1)^{p-1}\nabla\cdot\left(u(u+1)^{m_{2}-1}\nabla
v\right)$
$\displaystyle+\xi\int_{\Omega}(u+1)^{p-1}\nabla\cdot\left(u(u+1)^{m_{3}-1}\nabla
w\right)+\lambda\int_{\Omega}(u+1)^{p-1}u-\mu\int_{\Omega}(u+1)^{p-1}u^{r}\quad\text{on
}(0,T_{max}).$ Due to the divergence theorem, the three first integrals on the
right hand side make that the above identity becomes
$\displaystyle\varphi^{\prime}(t)=$
$\displaystyle-(p-1)\int_{\Omega}(u+1)^{p+m_{1}-3}\left\lvert\nabla
u\right\rvert^{2}+(p-1)\chi\int_{\Omega}u(u+1)^{p+m_{2}-3}\nabla u\cdot\nabla
v$ $\displaystyle-(p-1)\xi\int_{\Omega}u(u+1)^{p+m_{3}-3}\nabla u\cdot\nabla
w+\lambda\int_{\Omega}(u+1)^{p-1}u-\mu\int_{\Omega}(u+1)^{p-1}u^{r}\quad\text{for
all }t\in(0,T_{max}).$ By recalling (17), we rewrite the previous expression
as $\displaystyle\varphi^{\prime}(t)=$
$\displaystyle-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+(p-1)\chi\int_{\Omega}\nabla
F_{m_{2}}(u)\cdot\nabla v-(p-1)\xi\int_{\Omega}\nabla F_{m_{3}}(u)\cdot\nabla
w$
$\displaystyle+\lambda\int_{\Omega}(u+1)^{p-1}u-\mu\int_{\Omega}(u+1)^{p-1}u^{r}\quad\text{on
$(0,T_{max})$},$ and again the divergence theorem provides
$\displaystyle\varphi^{\prime}(t)=$
$\displaystyle-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}-(p-1)\chi\int_{\Omega}F_{m_{2}}(u)\Delta
v+(p-1)\xi\int_{\Omega}F_{m_{3}}(u)\Delta w$
$\displaystyle+\lambda\int_{\Omega}(u+1)^{p-1}u-\mu\int_{\Omega}(u+1)^{p-1}u^{r},\quad\text{for
all $t\in(0,T_{max})$}.$

The inequality

$\frac{1}{2^{r}}\int_{\Omega}(u+1)^{p-1+r}\leq\int_{\Omega}(u+1)^{p-1}u^{r}+\int_{\Omega}(u+1)^{p-1}\quad\text{on
}(0,T_{max}),$

justified by inequality (14), leads to the further bound

(19)
$\begin{split}\varphi^{\prime}(t)\leq&-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}-(p-1)\chi\int_{\Omega}F_{m_{2}}(u)\Delta
v+(p-1)\xi\int_{\Omega}F_{m_{3}}(u)\Delta w\\\
&+\lambda\int_{\Omega}(u+1)^{p}+\mu\int_{\Omega}(u+1)^{p-1}-\frac{\mu}{2^{r}}\int_{\Omega}(u+1)^{p-1+r}\quad\text{for
all }t\in(0,T_{max}).\end{split}$

Finally, a double application of Young’s inequality (recalling $r>1$) gives

$\lambda\int_{\Omega}(u+1)^{p}+\mu\int_{\Omega}(u+1)^{p-1}\leq\frac{\mu}{2^{r+1}}\int_{\Omega}(u+1)^{p-1+r}+c\textsubscript{30}\quad\text{on
}(0,T_{max}),$

which, in conjunction with inequality (19), leads to (18). ∎

From now on we will distinguish the analyses for the local problem and the
nonlocal one.

###  5.1. Study of the local problem ($\mathcal{L}$)

Let us separate the elliptic and parabolic cases, i.e., $\tau=0$ and $\tau=1$
respectively.

####  5.1.1. The parabolic-elliptic case: $\tau=0$

######  Lemma 5.4.

For the value of $\bar{p}>1$ found in Lemma 3.3, let conditions (5),
($\mathcal{A}_{1}$), ($\mathcal{A}_{1}$) and ($\mathcal{A}_{1}$) be satisfied.
Then $u\in L^{\infty}((0,T_{max}),L^{p}(\Omega))$ for all $p>\bar{p}$.

###### Proof.

The second and third equations in ($\mathcal{L}$) allow the substitution of
$\Delta v$ and $\Delta w$ in (18), obtaining once nonpositive terms are
dropped

(20)
$\begin{split}\varphi^{\prime}(t)\leq&-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}-(p-1)\chi\int_{\Omega}F_{m_{2}}(u)\left(\beta
v-f(u)\right)+(p-1)\xi\int_{\Omega}F_{m_{3}}(u)\left(\delta w-g(u)\right)\\\
&\quad-c\textsubscript{28}\int_{\Omega}(u+1)^{p+r-1}+c\textsubscript{29}\\\
\leq&-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+(p-1)\chi\int_{\Omega}F_{m_{2}}(u)f(u)+c\textsubscript{31}\int_{\Omega}F_{m_{3}}(u)w-(p-1)\xi\int_{\Omega}F_{m_{3}}(u)g(u)\\\
&\quad-c\textsubscript{28}\int_{\Omega}(u+1)^{p+r-1}+c\textsubscript{29}\quad\text{on
}(0,T_{max}).\end{split}$

With some calculations, by using the definition in (17), we find that

(21) $0\leq\frac{1}{p+j-1}u^{p+j-1}\leq
F_{j}(u)\leq\frac{1}{p+j-1}\left((u+1)^{p+j-1}-1\right).$

By considering (5), (20), and (21), we have

(22)
$\begin{split}\varphi^{\prime}(t)\leq&-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{32}\int_{\Omega}(u+1)^{p+m_{2}+k-1}+c\textsubscript{33}\int_{\Omega}(u+1)^{p+m_{3}-1}w\\\
&-c\textsubscript{34}\int_{\Omega}u^{p+m_{3}-1}(u+1)^{l}-c\textsubscript{28}\int_{\Omega}(u+1)^{p-1+r}+c\textsubscript{29}\\\
\leq&-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{32}\int_{\Omega}(u+1)^{p+m_{2}+k-1}+c\textsubscript{33}\int_{\Omega}(u+1)^{p+m_{3}-1}w\\\
&-c\textsubscript{35}\int_{\Omega}(u+1)^{p+m_{3}+l-1}+c\textsubscript{34}\int_{\Omega}(u+1)^{l}-c\textsubscript{28}\int_{\Omega}(u+1)^{p-1+r}+c\textsubscript{29}\quad\text{for
all }t\in(0,T_{max}),\end{split}$

where we have rearranged the term proportional to
$-\int_{\Omega}u^{p+m_{3}-1}(u+1)^{l}$ through (14). Let us deal with the two
terms $\int_{\Omega}(u+1)^{l}$ and $\int_{\Omega}(u+1)^{p+m_{3}-1}w$. For
every $l>0$, we have thanks to the Young inequality that for all $\hat{c}>0$

(23)
$\hat{c}\int_{\Omega}(u+1)^{l}\leq\frac{c\textsubscript{28}}{2}\int_{\Omega}(u+1)^{p-1+r}+c\textsubscript{36}\quad\text{on
}(0,T_{max}),$

(for $l\leq 1$ the boundedness of the mass (see again Lemma 5.1) would provide
a sharper estimate), whereas through (13d) applied with
$\varepsilon_{2}<c\textsubscript{35}$, from (22) we derive for all
$t\in(0,T_{max})$

(24)
$\varphi^{\prime}(t)\leq-c\textsubscript{37}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{32}\int_{\Omega}(u+1)^{p+m_{2}+k-1}-c\textsubscript{38}\int_{\Omega}(u+1)^{p+m_{3}+l-1}-\frac{c\textsubscript{28}}{2}\int_{\Omega}(u+1)^{p-1+r}+c\textsubscript{39}.$

Now we use assumption ($\mathcal{A}_{1}$) to write by Young’s inequality

$c\textsubscript{32}\int_{\Omega}(u+1)^{p+m_{2}+k-1}\leq
c\textsubscript{38}\int_{\Omega}(u+1)^{p+m_{3}+l-1}+c\textsubscript{40}\quad\text{on
}(0,T_{max}),$

which, introduced into (24), provides after neglecting the term from the
logistic source

$\varphi^{\prime}(t)\leq-c\textsubscript{37}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{41}\quad\text{for
all }t\in(0,T_{max}).$

The same estimate, up to constants, is obtained with assumption
($\mathcal{A}_{1}$) by controlling $\int_{\Omega}(u+1)^{p+m_{2}+k-1}$ with
$\int_{\Omega}(u+1)^{p-1+r}$, invoking the Young inequality, or relying on
assumption ($\mathcal{A}_{1}$) by exploiting (13a).

Finally, inequality (13e) yields the initial problem

$\begin{dcases}\varphi^{\prime}(t)\leq
c\textsubscript{42}-c\textsubscript{43}\varphi(t)^{\frac{p+m_{1}-1}{p\theta_{1}}}&\text{on
}(0,T_{max}),\\\
\varphi(0)=\frac{1}{p}\int_{\Omega}(u_{0}+1)^{p},\end{dcases}$

and ODI comparisons arguments ensure
$\varphi(t)\leq\max{\left\\{\varphi(0),\left(\frac{c\textsubscript{42}}{c\textsubscript{43}}\right)^{\frac{p\theta_{1}}{p+m_{1}-1}}\right\\}}$
for all $t\in(0,T_{max})$. ∎

####  5.1.2. The parabolic-parabolic case: $\tau=1$

######  Lemma 5.5.

For the value of $\bar{p}>1$ found in Lemma 3.3, let conditions (5),
($\mathcal{A}_{1}$), ($\mathcal{A}_{1}$), ($\mathcal{A}_{1}$) and
($\mathcal{A}_{1}$) be valid. Then $u\in
L^{\infty}((0,T_{max}),L^{p}(\Omega))$ for all $p>\bar{p}$.

###### Proof.

We start dealing with bounds for the terms
$\displaystyle\int_{\Omega}F_{m_{2}}(u)\Delta v$ and
$\displaystyle\int_{\Omega}F_{m_{3}}(u)\Delta w$. Relations in (21) and
Young’s inequality provide for all $t\in(0,T_{max})$

(25) $-(p-1)\chi\int_{\Omega}F_{m_{2}}(u)\Delta
v\leq(p-1)\chi\int_{\Omega}F_{m_{2}}(u)\left\lvert\Delta v\right\rvert\leq
c\textsubscript{44}\int_{\Omega}(u+1)^{p+m_{2}-1}\left\lvert\Delta
v\right\rvert\leq\int_{\Omega}(u+1)^{p+m_{2}+k-1}+c\textsubscript{45}\int_{\Omega}\left\lvert\Delta
v\right\rvert^{\frac{p+m_{2}-1+k}{k}}$

and analogously

(26) $(p-1)\xi\int_{\Omega}F_{m_{3}}(u)\Delta
w\leq\int_{\Omega}(u+1)^{p+m_{3}+l-1}+c\textsubscript{46}\int_{\Omega}\left\lvert\Delta
w\right\rvert^{\frac{p+m_{3}-1+l}{l}}.$

For $z=v$, $\eta=\beta$, $\psi=f(u)$ (and using (5)) and
$q=\frac{p+m_{2}+k-1}{k}$, inequality (9) allows us to write

(27) $\int_{0}^{t}e^{s}\int_{\Omega}\left\lvert\Delta
v\right\rvert^{\frac{p+m_{2}+k-1}{k}}\,ds\leq
C_{P}\left(1+\int_{0}^{t}e^{s}\int_{\Omega}(u+1)^{p+m_{2}+k-1}\,ds\right)\quad\text{on
}(0,T_{max}),$

and, reasoning similarly for the third equation in model ($\mathcal{L}$),

(28) $\int_{0}^{t}e^{s}\int_{\Omega}\left\lvert\Delta
w\right\rvert^{\frac{p+m_{3}+l-1}{l}}\,ds\leq\tilde{C}_{P}\left(1+\int_{0}^{t}e^{s}\int_{\Omega}(u+1)^{p+m_{3}+l-1}\,ds\right)\quad\text{for
all }t\in(0,T_{max}).$

By substituting (25) and (26) into (18), we obtain

(29)
$\begin{split}\varphi^{\prime}(t)\leq&-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+\int_{\Omega}(u+1)^{p+m_{2}+k-1}+c\textsubscript{45}\int_{\Omega}\lvert\Delta
v\rvert^{\frac{p+m_{2}+k-1}{k}}+\int_{\Omega}(u+1)^{p+m_{3}+l-1}\\\
&+c\textsubscript{46}\int_{\Omega}\lvert\Delta
w\rvert^{\frac{p+m_{3}+l-1}{l}}-c\textsubscript{28}\int_{\Omega}(u+1)^{p+r-1}+c\textsubscript{29}\quad\text{on
}(0,T_{max}).\end{split}$

Now we add to both sides of (29) the term $\varphi(t)$, and successively we
multiply by $e^{t}$. Since
$e^{t}\varphi^{\prime}(t)+e^{t}\varphi(t)=\frac{d}{dt}(e^{t}\varphi(t))$, an
integration over $(0,t)$ provides for all $t\in(0,T_{max})$

(30)
$\begin{split}e^{t}\varphi(t)\leq&\varphi(0)+\int_{0}^{t}e^{s}\left(-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+\int_{\Omega}(u+1)^{p+m_{2}+k-1}+c\textsubscript{45}\int_{\Omega}\lvert\Delta
v\rvert^{\frac{p+m_{2}+k-1}{k}}\right.\\\
&\left.+\int_{\Omega}(u+1)^{p+m_{3}+l-1}+c\textsubscript{46}\int_{\Omega}\lvert\Delta
w\rvert^{\frac{p+m_{3}+l-1}{l}}+\frac{1}{p}\int_{\Omega}(u+1)^{p}-c\textsubscript{28}\int_{\Omega}(u+1)^{p+r-1}+c\textsubscript{29}\right)\,ds.\end{split}$

Relations (27) and (28), in conjunction with Young’s inequality (recall $r>1$)
entail on $(0,T_{max})$ once inserted into (30)

(31) $\begin{split}\frac{e^{t}}{p}\int_{\Omega}u^{p}\leq
e^{t}\varphi(t)\leq&\varphi(0)+\int_{0}^{t}e^{s}\left(-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{47}\int_{\Omega}(u+1)^{p+m_{2}+k-1}\right.\\\
&\left.+c\textsubscript{48}\int_{\Omega}(u+1)^{p+m_{3}+l-1}-\frac{c\textsubscript{28}}{2}\int_{\Omega}(u+1)^{p+r-1}+c\textsubscript{49}\right)\,ds.\end{split}$

At this stage, analogously to what we have done in the previous lemma, if
restriction ($\mathcal{A}_{1}$) is supported with one between
($\mathcal{A}_{1}$) and ($\mathcal{A}_{1}$), Young’s inequality and (13b) make
that bound (31) is turned into

(32) $e^{t}\int_{\Omega}u^{p}\leq
c\textsubscript{50}+c\textsubscript{51}(e^{t}-1)\quad\textrm{for all
}t\in(0,T_{max}).$

The same procedure applies if we take into account ($\mathcal{A}_{1}$),
implying (13a), with either ($\mathcal{A}_{1}$) (by means of Young’s
inequality) or ($\mathcal{A}_{1}$), ensuring (13b).

Henceforth, also in this case inequality (32) is derived and as a consequence,
in each of the above situations, we conclude that

$\int_{\Omega}u^{p}\leq c\textsubscript{52}\quad\textrm{on }(0,T_{max}).$

∎

###  5.2. Study of the nonlocal problem ($\mathcal{N}$)

######  Lemma 5.6.

Let the hypotheses of Lemma 5.4 be valid. Then $u\in
L^{\infty}((0,T_{max}),L^{p}(\Omega))$ for all $p>\bar{p}$.

###### Proof.

Starting from relation (18), the second and the third equations in model
($\mathcal{N}$) lead to

$\begin{split}\varphi^{\prime}(t)\leq&-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+(p-1)\chi\int_{\Omega}F_{m_{2}}(u)f(u)+\frac{(p-1)\xi}{|\Omega|}\int_{\Omega}F_{m_{3}}(u)\int_{\Omega}g(u)\\\
&-(p-1)\xi\int_{\Omega}F_{m_{3}}(u)g(u)-c\textsubscript{28}\int_{\Omega}(u+1)^{p-1+r}+c\textsubscript{29}\\\
\leq&-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{53}\int_{\Omega}(u+1)^{p+m_{2}+k-1}+c\textsubscript{54}\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{l}\\\
&-c\textsubscript{55}\int_{\Omega}u^{p+m_{3}-1}(u+1)^{l}-c\textsubscript{28}\int_{\Omega}(u+1)^{p-1+r}+c\textsubscript{29}\quad\text{for
all }t\in(0,T_{max}),\end{split}$

where we have taken in mind the properties of $F_{j}$ and $f$ and $g$ fixed in
(5) and (21), and dropped nonpositive terms. In turn, thanks again to (14), we
have

$\displaystyle\varphi^{\prime}(t)$
$\displaystyle\leq-\frac{4(p-1)}{(p+m_{1}-1)^{2}}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{53}\int_{\Omega}(u+1)^{p+m_{2}+k-1}+c\textsubscript{54}\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{l}$
$\displaystyle\phantom{\leq}-c\textsubscript{56}\int_{\Omega}(u+1)^{p+m_{3}+l-1}+c\textsubscript{57}\int_{\Omega}(u+1)^{l}-c\textsubscript{28}\int_{\Omega}(u+1)^{p-1+r}+c\textsubscript{29}\quad\text{on
}(0,T_{max}).$

With the aid of estimate (23), by exploiting (13c) with
$\varepsilon_{1}<c\textsubscript{56}$, we obtain on $(0,T_{max})$

$\varphi^{\prime}(t)\leq-c\textsubscript{11}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{53}\int_{\Omega}(u+1)^{p+m_{2}+k-1}-c\textsubscript{58}\int_{\Omega}(u+1)^{p+m_{3}+l-1}-\frac{c\textsubscript{28}}{2}\int_{\Omega}(u+1)^{p-1+r}+c\textsubscript{59}.$

Since up to constants the previous estimate coincides with (24), we can follow
the same arguments of Lemma 5.4 to conclude. ∎

##  6\. Proof of Theorems 2.2 and 2.3

For $\tau=0$, we use respectively Lemma 5.4 and Lemma 5.6, supported by the
extension criterion in Lemma 4.1; in this way Theorem 2.2 is established.
Theorem 2.3 is evidently obtained by invoking Lemma 5.5.

#### Acknowledgements

All the authors are partially supported by the research project Analysis of
PDEs in connection with real phenomena, CUP F73C22001130007, funded by
Fondazione di Sardegna, annuity 2021. AC and SF are members of the Gruppo
Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). RDF is member
of the Gruppo Nazionale per il Calcolo Scientifico (GNCS) of INdAM and
acknowledges financial support by PNRR e.INS Ecosystem of Innovation for Next
Generation Sardinia (CUP F53C22000430001, codice MUR ECS0000038). SF is also
partially supported by: Research project of MIUR (Italian Ministry of
Education, University and Research) Prin 2022 Nonlinear differential problems
with applications to real phenomena (Grant Number: 2022ZXZTN2).

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