linecolor=gray,linewidth=0.8pt figurem

# Properties of given and detected unbounded solutions to a class of
chemotaxis models

###### Abstract.

This paper deals with unbounded solutions to a class of chemotaxis systems. In
particular, for a rather general attraction-repulsion model, with nonlinear
productions, diffusion, sensitivities and logistic term, we detect Lebesgue
spaces where given unbounded solutions blow-up also in the corresponding norms
of those spaces; subsequently, estimates for the blow-up time are established.
Finally, for a simplified version of the model, some blow-up criteria are
proved.

More precisely, we analyze a zero-flux chemotaxis system essentially described
as

($\Diamond$) $\begin{dcases}u_{t}=\nabla\cdot((u+1)^{m_{1}-1}\nabla u-\chi
u(u+1)^{m_{2}-1}\nabla v+\xi u(u+1)^{m_{3}-1}\nabla w)+\lambda u-\mu
u^{k}&\text{ in }\Omega\times(0,T_{max}),\\\ 0=\Delta
v-\frac{1}{\lvert\Omega\rvert}\int_{\Omega}u^{\alpha}+u^{\alpha}=\Delta
w-\frac{1}{\lvert\Omega\rvert}\int_{\Omega}u^{\beta}+u^{\beta}&\text{ in
}\Omega\times(0,T_{max}).\end{dcases}$

The problem is formulated in a bounded and smooth domain $\Omega$ of
$\mathbb{R}^{n}$, with $n\geq 1$, for some $m_{1},m_{2},m_{3}\in\mathbb{R}$,
$\chi,\xi,\alpha,\beta,\lambda,\mu>0$, $k>1$, and with $T_{max}\in(0,\infty]$.
A sufficiently regular initial data $u_{0}\geq 0$ is also fixed.

Under specific relations involving the above parameters, one of these always
requiring some largeness conditions on $m_{2}+\alpha$,

  1. i)

we prove that any given solution to ($\Diamond$ ‣ Properties of given and
detected unbounded solutions to a class of chemotaxis models), blowing-up at
some finite time $T_{max}$, becomes unbounded also in
$L^{\mathfrak{p}}(\Omega)$-norm, for all
${\mathfrak{p}}>\frac{n}{2}\left(m_{2}-m_{1}+\alpha\right)$;

  2. ii)

we give lower bounds $T$ (depending on $\int_{\Omega}u_{0}^{\bar{p}}$) of
$T_{max}$ for the aforementioned solutions in some $L^{\bar{p}}(\Omega)$-norm,
being $\bar{p}=\bar{p}(n,m_{1},m_{2},m_{3},\alpha,\beta)\geq\mathfrak{p}$;

  3. iii)

whenever $m_{2}=m_{3}$, we establish sufficient conditions on the parameters
ensuring that for some $u_{0}$ solutions to ($\Diamond$ ‣ Properties of given
and detected unbounded solutions to a class of chemotaxis models) effectively
are unbounded at some finite time.

Within the context of blow-up phenomena connected to problem ($\Diamond$ ‣
Properties of given and detected unbounded solutions to a class of chemotaxis
models), this research partially improves the analysis in [WZZ23] and moreover
contributes to enrich the level of knowledge on the topic.

###### Key words and phrases:

Chemotaxis, Attraction-repulsion, Nonlinear production, Blow-up time, Lower
bound.
⋆Corresponding author<EMAIL_ADDRESS>
###### 2020 Mathematics Subject Classification:

Primary: 35B44, 35K55, 35Q92. Secondary: 92C17.

Alessandro Columbu, Silvia Frassu⋆ Giuseppe Viglialoro

Dipartimento di Matematica e Informatica

Università di Cagliari

Via Ospedale 72, 09124. Cagliari (Italy)

##  1\. Introduction, motivations and state of the art

###  1.1. The continuity equation; the initial-boundary value problem

The well known continuity equation

(1) $u_{t}=\nabla\cdot F+h$

describes the transport of some quantity $u=u(x,t)$, at the position $x$ and
at the time $t>0$. In this equation the flux $F$ models the motion of such a
quantity, whereas $h$ is an additional source idealizing some external action
by means of which $u$ itself may be created or destroyed throughout the time.

In this paper we are interested in the analysis of equation (1) in the context
of self organization mechanisms for biological populations, i.e. phenomena for
which organisms or entities direct their trajectory in response to one or more
chemical stimuli. More precisely, we want to deal with the motion of a certain
cell density $u=u(x,t)$ whose flux has a smooth diffusive part and another
contrasting this spread. In the specific, this counterpart accounts of an
attractive and a repulsive effect, associated to two chemical signals and
indicated respectively with $v=v(x,t)$ and $w=w(x,t)$; $v$ (the
chemoattractant) tends to gather the cells, $w$ (the chemorepellent) to
scatter them. Additionally an external source with an increasing and
decreasing effect on the cell density is also included.

For our purposes, wanting to formulate what said above in terms of the
continuity equation (1), it appears meaningful (and convenient) defining for
$\chi,\xi,\lambda,\mu>0$, $m_{1},m_{2},m_{3}\in\mathbb{R}$ and $k>1$ the
fluxes $F=F_{m_{1},m_{2},m_{3}}=F_{m_{1},m_{2},m_{3}}(u,v,w)$,
$G_{m_{1}}=G_{m_{1}}(u)$, $H_{m_{2}}=H_{m_{2}}(u,v)$,
$I_{m_{3}}=I_{m_{3}}(u,w)$ and the source $h=h_{k}=h_{k}(u)$:

(2) $\begin{cases}F=F_{m_{1},m_{2},m_{3}}=(u+1)^{m_{1}-1}\nabla u-\chi
u(u+1)^{m_{2}-1}\nabla v+\xi u(u+1)^{m_{3}-1}\nabla
w=:G_{m_{1}}+H_{m_{2}}+I_{m_{3}}\\\ h=h_{k}=\lambda u-\mu u^{k}.\end{cases}$

In this way, the diffusion is smoother and smoother for higher and higher
values of $m_{1}$, the aggregation/repulsion effects $-\chi
u(u+1)^{m_{2}-1}\nabla v/\xi u(u+1)^{m_{3}-1}\nabla w$ increase for larger
sizes of $\chi$ and $\xi$ and $m_{2}$ and $m_{3}$, and the cell density may
increment with rate $\lambda u$ and may attenuate with rate $-\mu u^{k}$.
Naturally, since the flux is influenced by the two signals $v$ and $w$, we
will have to consider two more equations (which for the time being we indicate
with $P(v)=0$ and $Q(w)=0$, but that will be specified later), one for the
chemoattractant $v$ and another for the chemorepellent $w$, to be coupled with
the continuity equation. Furthermore, the analysis is studied in impenetrable
domains (so homogeneous Neumann or zero flux boundary conditions are imposed)
and some initial configurations for the cell and chemical densities are
assigned: essentially, with position (2) in mind, we are concerned with this
initial boundary value problem:

(3) $\begin{cases}u_{t}=\nabla\cdot F_{m_{1},m_{2},m_{3}}+h_{k}&\textrm{ in
}\Omega\times(0,T_{max}),\\\ P(v)=Q(w)=0&\textrm{ in
}\Omega\times(0,T_{max}),\\\ u_{0}(x)=u(x,0)\geq 0;\;v_{0}(x)=v(x,0)\geq
0;\;w_{0}(x)=w(x,0)\geq 0&x\in\bar{\Omega},\\\
u_{\nu}=v_{\nu}=w_{\nu}=0&\textrm{ on
}\partial\Omega\times(0,T_{max}).\end{cases}$

The problem is formulated in a bounded and smooth domain $\Omega$ of
$\mathbb{R}^{n}$, with $n\geq 1$, $u_{\nu}$ (and similarly for $v_{\nu}$ and
$w_{\nu}$) indicates the outward normal derivative of $u$ on $\partial\Omega$.
Moreover, $T_{max}\in(0,\infty]$ identifies the maximum time up to which
solutions to the system can be extended.

###  1.2. A view on the state of the art: the attractive and the repulsive
models and the attraction-repulsion model

The aforementioned discussion finds, of course, its roots in the well-known
Keller–Segel models idealizing chemotaxis phenomena (see the celebrated papers
[KS70, KS71a, KS71b]), that since the last 50 years have been attracting the
interest of the mathematical community.

In particular, if we refer to chemotaxis models with single proliferation
signal, taking in mind (2), problem (3) is a (more general) combination of
this aggregative signal-production mechanism

(4) $u_{t}=\nabla\cdot\left(G_{1}+H_{1}\right)=\Delta
u-\chi\nabla\cdot(u\nabla v)\quad\textrm{and}\quad P(v)=P^{\tau}_{1}(v)=\tau
v_{t}-\Delta v+v-u=0,\quad\textrm{ in }\Omega\times(0,T_{max}),$

and this repulsive signal-production one

(5) $u_{t}=\nabla\cdot\left(G_{1}+I_{1}\right)=\Delta u+\xi\nabla\cdot(u\nabla
w)\quad\textrm{and}\quad Q(w)=Q^{\tau}_{1}(w)=\tau w_{t}-\Delta
w+w-u=0,\quad\textrm{ in }\Omega\times(0,T_{max}).$

(Here $\tau\in\\{0,1\\}$ and it distinguishes between a stationary and
evolutive equation for the chemical.) The above models present linear
diffusion and linear production rates; specifically, $v$ and $w$ are linearly
produced by the cells themselves, and their mechanism is opposite when in
$P^{\tau}_{1}(v)$ the term $v-u$ (or in $Q_{1}(w)$ the term $w-u$) is replaced
by $uv$ (or $uw$); in this case the particle density consumes the chemical.
(We will spend only few words on models with absorption.) As far as problem
(4) is concerned, since the attractive signal $v$ increases with $u$, the
natural spreading process of the cells’ density could interrupt and very high
and spatially concentrated spikes formations (chemotactic collapse or blow-up
at finite time) may appear; this is, generally, due to the size of the
chemosensitvity $\chi$, the initial mass of the particle distribution, i.e.,
$m=\int_{\Omega}u_{0}(x)dx,$ and the space dimension $n$. In this direction,
the reader interested in learning more can find in [HV97, JL92, Nag01, Win10]
analyses dealing with existence and properties of global, uniformly bounded,
or blow-up (local) solutions to models connected to (4).

On the other hand, for nonlinear segregation chemotaxis models like those we
are interested in, when in problem (4) one has that
$P(v)=P^{1}_{\alpha}(v)=v_{t}-\Delta v+v-u^{\alpha}=0$, with
$0<\alpha<\frac{2}{n}$ ($n\geq 1$), uniform boundedness of all its solutions
is proved in [LT16].

Concerning the literature about problem (5), it seems rather poor and general
(see, for instance, [Moc74, Moc75] for analyses on similar contexts). In
particular, no result on the blow-up scenario is available; this is meaningful
due to the repulsive nature of the phenomenon.

Contrarily, the level of understanding for attraction-repulsion chemotaxis
problems involving both (4) and (5) is sensitively rich; more specifically, if
we refer to the linear diffusion and sensitivities version of model (3), for
which $F=F_{1,1,1}$ and $P^{\tau}_{\alpha}(v)=\tau v_{t}-\Delta v+bv-
au^{\alpha}$ and $Q^{\tau}_{\beta}(w)=\tau w_{t}-\Delta w+dw-cu^{\beta}$,
$a,b,c,d,\alpha,\beta>0$, equipped with regular initial data $u_{0}(x),\tau
v_{0}(x),\tau w_{0}(x)\geq 0$, we can recollect the following outcomes. In the
absence of logistics ($h_{k}\equiv 0$), when linear growths of the
chemoattractant and the chemorepellent are taken into consideration, and for
elliptic equations for the chemicals (i.e. when
$P^{0}_{1}(v)=Q^{0}_{1}(w)=0$), the value $\Theta:=\chi a-\xi c$ measures the
difference between the attraction and repulsion impacts, and it is such that
whenever $\Theta<0$ (repulsion-dominated regime), in any dimension all
solutions to the model are globally bounded, whereas for $\Theta>0$
(attraction-dominated regime) and $n=2$ unbounded solutions can be detected
(see [GJZ18, LL16, TW13, Vig19, YGZ17] for some details on the issue). Indeed,
for more general expressions of the proliferation laws, modelled by the
equations $P^{0}_{\alpha}(v)=Q^{0}_{\beta}(w)=0$, to the best of our
knowledge, [Vig21] is the most recent result in this direction; herein some
interplay between $\alpha$ and $\beta$ and some technical conditions on $\xi$
and $u_{0}$ are established so to ensure globality and boundedness of
classical solutions. (See also [CY22] for blow-up results in the frame of
nonlinear attraction-repulsion models with logistics as those formulated in
(3) with $F=F_{m_{1},m_{2},m_{3}}$ and $h=h_{k}$, and with linear segregation
for the stimuli, i.e. with equations for $v,w$ reading as
$P_{1}^{0}(v)=Q_{1}^{0}(w)=0$.)

Putting our attention on evolutive equations for chemoattractant and
chemorepellent, $P^{1}_{1}(v)=Q^{1}_{1}(w)=0$, in [TW13] it is proved that in
two-dimensional domains sufficiently smooth initial data emanate global-in-
time bounded solutions whenever

$\Theta<0\quad\textrm{and}\quad
b=d\quad\textrm{or}\quad\Theta<0\quad\textrm{and}\quad-\frac{\chi^{2}\alpha^{2}(b-d)^{2}}{2\Theta{b}^{2}C}\int_{\Omega}u_{0}(x)dx\leq
1,\quad\textrm{for some }\;C>0.$

(As to blow-up results we are only aware of [Lan21], where unbounded solutions
in three-dimensional domains are constructed.) When $h=h_{k}\not\equiv 0$, for
both linear and nonlinear productions scenarios, and stationary or evolutive
equations (formally, $P^{\tau}_{\alpha}(v)=Q^{\tau}_{\beta}(w)=0$), criteria
toward boundedness, long time behaviors and blow-up issues for related
solutions are studied in [HTZ20, ZLZ22, CMTY21, RL22].

###  1.3. The nonlocal case

In this work we are mainly interested to the so-called nonlocal models tied to
(3), for which, specifically, $P(v)=P_{\alpha}(v):=\Delta
v-\frac{1}{|\Omega|}\int_{\Omega}u^{\alpha}+u^{\alpha}=0$, and
$Q(w)=Q_{\beta}(w):=\Delta
w-\frac{1}{|\Omega|}\int_{\Omega}u^{\beta}+u^{\beta}$. In particular,
recalling the position in (2), we herein mention the most recent researches we
are aware about and inspiring our study; to this purpose we refer to the only
attraction version

(6)
$u_{t}=\nabla\cdot\left(G_{m_{1}}+H_{m_{2}}\right)+h_{k}\quad\textrm{and}\quad
P_{\alpha}(v)=0\quad\textrm{ in }\Omega\times(0,T_{max}),$

and the attraction-repulsion one

(7) $u_{t}=\nabla\cdot F_{m_{1},m_{2},m_{3}}+h_{k}\quad\textrm{and}\quad
P_{\alpha}(v)=Q_{\beta}(w)=0\quad\textrm{ in }\Omega\times(0,T_{max}).$

For the linear diffusion and sensitivity version of model (6), i.e. with flux
$G_{1}+H_{1}$, if $h_{k}\equiv 0$, it is known that boundedness of solutions
is achieved for any $n\geq 1$ and $0<\alpha<\frac{2}{n}$, whereas for
$\alpha>\frac{2}{n}$ blow-up phenomena may be observed (see [Win18]). When
dampening logistic terms take part in the mechanism, for the linear flux
$G_{1}+H_{1}$ appearances of $\delta$-formations at finite time have been
detected for some sub-quadratic growth of $h_{k}$, and precisely for $h_{k}$
with $1<k<\frac{n(\alpha+1)}{n+2}<2$ (see [YMXD21]). But there is more; for
the limit linear production scenario, corresponding to $P_{1}(v)=0$, some
unbounded solutions have been constructed in [Fue21] even for quadratic
sources $h=h_{2}$, whenever $n\geq 5$ and
$\mu\in\left(0,\frac{n-4}{n}\right).$

In nonlinear models without dampening logistic effects (i.e., general flux
$G_{m_{1}}+H_{m_{2}}$ and $h_{k}\equiv 0$) and linear production (i.e.
$P_{1}(v)=0$), in [WD10] it is shown, among other things, that for $m_{1}\leq
1$, $m_{2}>0$, $m_{2}>m_{1}+\frac{2}{n}-1$ situations with unbounded solutions
at some finite time $T_{max}$ can be found. (See also [MNV20] for questions
connected to estimates of $T_{max}$.) Some results have been also extended in
[Tan22] when $h_{k}\not\equiv 0$ and for nonlinear segregation contexts,
$P_{\alpha}(v)=0$; in particular, inter alia, for $m_{1}\in\mathbb{R}$,
$m_{2}>0$, blow-up phenomena are seen to appear if
$m_{2}+\alpha>\max\bigl{\\{}m_{1}+\frac{2}{n}k,k\bigr{\\}}$, whenever
$m_{1}\geq 0$ or $m_{2}+\alpha>\max\bigl{\\{}\frac{2}{n}k,k\bigr{\\}}$,
provided $m_{1}<0$.

On the other hand, for the attraction-repulsion models, in [LL21] it is
proved, together with other results, that if $P_{\alpha}(v)=Q_{\beta}(w)=0$,
$\alpha>\frac{2}{n}$ and $\alpha>\beta$ ensure the existence of unbounded
solutions to (7) for the linear flux $F=F_{1,1,1}$, without logistic
($h_{k}\equiv 0$). Conversely, detecting gathering mechanisms for the
nonlinear situation is more complex and we are only aware of [WZZ23]; indeed
this issue is therein addressed only for $F=F_{m_{1},1,1}$, with
$m_{1}\in\mathbb{R}$, but even in presence of dampening logistics. Since in
our research we will show the existence of blow-up solutions for a larger
class of fluxes, precisely for $F=F_{m_{1},m_{2},m_{2}}$ with
$m_{1}\in\mathbb{R}$ and any $m_{2}=m_{3}>0$, we will spend more words to
analyze details of [WZZ23] below, precisely in $\S$3.2.

##  2\. Presentation of the model and of the main results. Aims of the paper

###  2.1. The formulation of the mathematical problem

In light of what we presented so far, and under the aforementioned main
positions, basically in this paper we are interested in properties of
unbounded classical solutions $(u,v,w)=(u(x,t),v(x,t),w(x,t))$ to this problem

(8) $\begin{dcases}u_{t}=\nabla\cdot((u+1)^{m_{1}-1}\nabla u-\chi
u(u+1)^{m_{2}-1}\nabla v+\xi u(u+1)^{m_{3}-1}\nabla w)+\lambda u-\mu
u^{k}&\text{ in }\Omega\times(0,T_{max}),\\\ 0=\Delta
v-m_{1}(t)+f_{1}(u)&\text{ in }\Omega\times(0,T_{max}),\\\ 0=\Delta
w-m_{2}(t)+f_{2}(u)&\text{ in }\Omega\times(0,T_{max}),\\\
u_{\nu}=v_{\nu}=w_{\nu}=0&\text{ on }\partial\Omega\times(0,T_{max}),\\\
u(x,0)=u_{0}(x)&x\in\bar{\Omega},\\\
\int_{\Omega}v(x,t)dx=\int_{\Omega}w(x,t)dx=0&\text{ for all
}t\in(0,T_{max}).\end{dcases}$

Additionally, the initial cell distribution $u_{0}:=u_{0}(x)$ and the
production laws $f_{i}=f_{i}(u)$ (for $i\in\\{1,2\\}$) are supposed to be
nonnegative and sufficiently regular and the functions $m_{i}(t)$ are defined
for compatibility in the second and third equation (by integrating these over
$\Omega$) in terms of $f_{i}(u)$ themselves, exactly as

(9)
$m_{1}(t)=\frac{1}{\lvert\Omega\rvert}\int_{\Omega}f_{1}(u)\quad\textrm{and}\quad
m_{2}(t)=\frac{1}{\lvert\Omega\rvert}\int_{\Omega}f_{2}(u).$

Herein we assume

(10) $0\leq
f_{i}\in\bigcup_{\theta\in(0,1)}C^{\theta}_{loc}\left([0,\infty)\right)\cap
C^{1}((0,\infty))\;\;\text{ and }0\leq
u_{0}\in\bigcup_{\theta\in(0,1)}C^{\theta}(\bar{\Omega});$

we also might need that for all $s\geq 0$, $\alpha,\beta>0$ and some
$k_{1},k_{2},k_{3}>0$,

(11) $f_{1}(s)\leq k_{1}(s+1)^{\alpha}\quad\textrm{and}\quad f_{2}(s)\leq
k_{2}(s+1)^{\beta},\quad$

or

(12) $f_{1},f_{2}\,\text{ nondecreasing},f_{1}(s)\geq
k_{3}(s+1)^{\alpha},\quad f_{2}(s)\leq k_{2}(s+1)^{\beta}\textrm{ and
$u_{0}=u_{0}(|x|)$ radially symmetric and nonincreasing}.$

######  Remark 1.

Let us clarify that in nonlocal models, and henceforth in this work, $v$
stands for the deviation of the chemoattractant; the deviation is the
difference between the signal concentration and its mean, and that it changes
sign in contrast to what happens with the cell and signal densities (which are
nonnegative). In particular, it follows from the definition of $v$ itself that
its mean is zero (as specified in the last positions of the problem (8)),
which in turn ensures the uniqueness of the solution of the Poisson equation
under homogeneous Neumann boundary conditions. The same comments apply for the
chemorepellent $w$. (We did not introduce different symbols to indicate the
chemicals and their deviations since it is clear from the context.)

###  2.2. Presentation of the Theorems. Overall aims of the paper

Our project finds its motivations in the observation that there is no
automatic connection between the occurrence of blow-up for solutions to model
(8) in the $L^{\infty}(\Omega)$-norm and that in $L^{p}(\Omega)$-norm ($p>1$).
Indeed, for a bounded domain $\Omega$, it is seen that

$\|u(\cdot,t)\|_{L^{p}(\Omega)}\leq|\Omega|^{\frac{1}{p}}\|u(\cdot,t)\|_{L^{\infty}(\Omega)},$

so that unboundedness in $L^{p}(\Omega)$-norm implies that in
$L^{\infty}(\Omega)$-norm, but oppositely $\int_{\Omega}u^{p}$ might even
remain bounded in a neighborhood of $T_{max}$ when $\max_{\Omega}u$
uncontrollably increases at some finite time $T_{max}$.

In light of this, in order to bridge the gap between the analysis of the blow-
up time $T_{max}$ in the two different mentioned norms, we aim at

  1. (i)

detecting suitable $L^{p}(\Omega)$ spaces, for certain $p$ depending on
$n,m_{1},m_{2},m_{3},\alpha$ and $\beta$, such that given unbounded solutions
also blow up in the associated $L^{p}(\Omega)$-norms;

  2. (ii)

providing lower bounds for the blow-up time of the aforementioned solutions in
these $L^{p}(\Omega)$-norms.

Additionally, another objective of our work is

  1. (iii)

giving sufficient conditions on the data of the model such that related
solutions are actually unbounded at finite time.

In order to deal with issues (i), (ii) and (iii), we fix the following
relations, determining some precise interplay involving constants defining
system (8):

######  Assumptions 2.1.

Let $n\in\mathbb{N}$, $m_{1},m_{2},m_{3}\in\mathbb{R}$ and
$\alpha,\beta,\chi,\xi,\lambda,\mu>0$ and $k>1$ be such that

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

$m_{2}+\alpha>m_{1}+\frac{2}{n}$.

Moreover, for $\beta>0$, let either

  1. ($\mathcal{H}_{2}$)

$m_{2}+\alpha>\max\\{1,m_{3}+\beta\\}$ or $m_{2}+\alpha\geq m_{3}+\beta$ and
$m_{1}>1-\frac{2}{n}$,

whereas, for $\beta\in(0,1]$, let either

  1. ($\mathcal{H}_{3}$)

$m_{3}\leq 1$ and $m_{1}>1-\frac{2}{n}$ or $m_{2}+\alpha\geq m_{3}$ and
$m_{1}>1-\frac{2}{n}$.

Finally, for $m_{2},\beta>0$, let also

  1. ($\mathcal{H}_{4}$)

$\alpha>\beta\quad\text{and}\quad\begin{cases}m_{2}+\alpha>\max\bigl{\\{}m_{1}+\frac{2}{n}k,k\bigr{\\}}&\text{
if }m_{1}\geq 0,\\\ m_{2}+\alpha>\max\bigl{\\{}\frac{2}{n}k,k\bigr{\\}}&\text{
if }m_{1}<0.\end{cases}$

With the support of the above position, and as far as the analysis of (i) is
concerned, in the spirit of [Fre18, Theorem 2.2] we will prove this first
result, dealing with properties of given unbounded solutions to model (8).
Specifically, the proof is based on the analysis of the functional
$\varphi(t)=\frac{1}{p}\int_{\Omega}(u+1)^{p}$, defined for local solutions to
problem (8) on $(0,T_{max})$. We will show that if for some $p$ sufficiently
large $\varphi(t)$ is uniformly bounded in time, then it also is for any
arbitrarily large $p>1$, so contrasting with the unboundedness of $u$ itself.

######  Theorem 2.2.

Let $\Omega$ be a bounded and smooth domain of $\mathbb{R}^{n}$, and condition
($\mathcal{H}_{1}$) as well as one of
($\mathcal{H}_{2}$),($\mathcal{H}_{2}$),($\mathcal{H}_{3}$),
($\mathcal{H}_{3}$) in Assumptions 2.1 hold true. Moreover, for $f_{i}$ and
$u_{0}$ complying with (10) and (11), let $(u,v,w)$, with

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

be a solution to problem (8) which blows-up at some finite time $T_{max}$, in
the sense that

(13) $\limsup_{t\to T_{max}}{\lVert
u(\cdot,t)\rVert_{L^{\infty}(\Omega)}}=+\infty.$

Then, for any $\mathfrak{p}>\frac{n}{2}\left(m_{2}-m_{1}+\alpha\right)$, we
also have

$\limsup_{t\to T_{max}}{\lVert
u(\cdot,t)\rVert_{L^{\mathfrak{p}}(\Omega)}}=+\infty.$

Successively aim (ii) is achieved by establishing for the same functional
$\varphi(t)$ a first order differential inequality (ODI) of the type
$\varphi^{\prime}(t)\leq\Psi(\varphi(t))$ on $(0,T_{max})$. In particular, for
any $\tau_{0}>0$ the function $\Psi(\tau)$ obeys the Osgood criterion
([Osg98]),

(14) $\int_{\tau_{0}}^{\infty}\frac{d\tau}{\Psi(\tau)}<\infty\quad\textrm{with
}\tau_{0}>0,$

so that an integration on $(0,T_{max})$ of the ODI implies, whenever
$\limsup_{t\rightarrow T_{max}}\varphi(t)=\infty$, the following lower bound
for $T_{max}$

$T_{max}\geq\int_{\varphi(0)}^{\infty}\frac{d\varphi}{\Psi(\varphi)}:=T,$

and thereby a safe interval of existence $[0,T)$ for solutions to the model
itself.

######  Theorem 2.3.

Let the hypotheses of Theorem 2.2 be satisfied. Then there exist $\bar{p}>1$
and positive constants $\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}$, as
well as $\gamma>\delta>1$, such that the blow-up time $T_{max}$ complies with
both the implicit estimate

(15)
$T_{max}\geq\int_{\varphi(0)}^{\infty}\frac{d\tau}{\mathcal{A}\tau^{\gamma}+\mathcal{B}\tau^{\delta}+\mathcal{C}},$

and the explicit one

$T_{max}\geq\frac{\varphi(0)^{1-\gamma}}{\mathcal{D}(\gamma-1)},$

where $\varphi(0)=\displaystyle\frac{1}{p}\int_{\Omega}(u_{0}+1)^{p}$, for any
$p\geq\bar{p}$.

Finally, the last theorem (connected to item (iii)) establishes (at least in a
particular case) the existence of unbounded solutions to system (8), so as to
make the two previous statements meaningful. The basic idea consists on the
analysis of the temporal evolution of the functional
$\phi(t):=\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)U(s,t)\,ds$ for
$t\in[0,T_{max})$, being $U(s,t)$ the so-called mass accumulation function of
$u$, obeying a superlinear ODI.

######  Theorem 2.4.

Let $\chi,\xi,\lambda,\mu,m_{2}=m_{3}>0$ and $k>1$. Additionally, for some
$R>0$, let $\Omega=B_{R}(0)\subset\mathbb{R}^{n}$ be a ball, $u_{0}$, $f_{1}$
and $f_{2}$ satisfy (10) and (12). Finally, let hypotheses ($\mathcal{H}_{4}$)
be satisfied. Then for any $M_{0}>C$, with
$C=\left(\frac{\lambda}{\mu}|\Omega|^{k-1}\right)^{\frac{1}{k-1}}$, there
exist $\epsilon_{0}\in(0,M_{0})$ and $r_{*}\in(0,R)$ with the property that
whenever $u_{0}$ is also chosen such that

(16)
$\int_{\Omega}u_{0}(x)dx=M_{0}\quad\text{and}\quad\int_{B_{r_{*}}(0)}u_{0}(x)dx\geq
M_{0}-\epsilon_{0},$

the corresponding classical solution $(u,v,w)$ to model (8) blows up at some
finite time $T_{max}$, in the sense of relation (13).

##  3\. Miscellaneous and general comments

###  3.1. On the parameters $\mathfrak{p}$ and $\bar{p}$

We herein want to spend some words on the role of the parameters
$\mathfrak{p}$ and $\bar{p}$ appearing in Theorems 2.2 and 2.3. In particular,
as we will observe in Lemma 4.2 below, $\mathfrak{p}$ and $\bar{p}$ depend on
$n,m_{1},m_{2},m_{3},\alpha$ and $\beta$, and Figure 1 collects some of their
values on the $p$ axis. In the specific, for a blowing-up solution to (8), we
will have that $\int_{\Omega}u^{p}$ is bounded on $(0,T_{max})$ for $p=1$,
while it blows up for $p\geq\mathfrak{p}$; in general, the behaviour of
$\int_{\Omega}u^{p}$ on $(0,T_{max})$ for $p\in\left(1,\mathfrak{p}\right)$ is
unknown. On the other hand, an estimate for $T_{max}$ is given in terms of
$\frac{1}{p}\int_{\Omega}\left(u_{0}+1\right)^{p}$, for $p\geq\bar{p}$, but
not when $p\in\left(\mathfrak{p},\bar{p}\right)$.

$\circ$$p$$\circ$$1$$\mathfrak{p}$$\bar{p}$

$\circ$$p$$\circ$$1$$\mathfrak{p}$$\bar{p}$

$\circ$$p$$\circ$$1$$\mathfrak{p}$$\bar{p}$

$\circ$$p$$\circ$$1$$\mathfrak{p}$$\bar{p}$

Figure 1. Some infimum of $\mathfrak{p}$ and $\bar{p}$, taken from their
definitions in Lemma 4.2. From top to bottom:  ➢ Under assumption
($\mathcal{H}_{2}$), and $n=1$, $m_{1}=\frac{81}{50}$,
$m_{2}=-\frac{149}{100}$, $m_{3}=\frac{33}{20}$, $\alpha=\frac{587}{100}$ and
$\beta=\frac{63}{25}$, we have
$\mathfrak{p}=\frac{n}{2}\left(m_{2}-m_{1}+\alpha\right)=\frac{69}{50}$ and
$\bar{p}=m_{3}(n+2)(n+1)=\frac{99}{10}$; ➢ Under the assumption
($\mathcal{H}_{2}$), and $n=5$, $m_{1}=-\frac{3}{50}$, $m_{2}=\frac{6}{5}$,
$m_{3}=-\frac{143}{100}$, $\alpha=\frac{163}{100}$ and
$\beta=\frac{169}{100}$, we have
$\mathfrak{p}=\frac{n}{2}\left(m_{2}-m_{1}+\alpha\right)=\frac{289}{40}$ and
$\bar{p}=m_{2}(n+2)(n+1)=\frac{252}{5}$; ➢ Under the assumption
($\mathcal{H}_{2}$), and $n=4$, $m_{1}=-\frac{187}{100}$,
$m_{2}=-\frac{89}{100}$, $m_{3}=-\frac{181}{100}$, $\alpha=\frac{353}{100}$
and $\beta=\frac{19}{50}$, we have
$\mathfrak{p}=\frac{n}{2}\left(m_{2}-m_{1}+\alpha\right)=\frac{451}{50}$ and
$\bar{p}=n\alpha=\frac{353}{25}$; ➢ Under the assumption ($\mathcal{H}_{2}$),
and $n=4$, $m_{1}=\frac{47}{50}$, $m_{2}=\frac{6}{25}$,
$m_{3}=-\frac{63}{50}$, $\alpha=\frac{179}{100}$ and $\beta=\frac{119}{50}$,
we have
$\mathfrak{p}=\frac{n}{2}\left(m_{2}-m_{1}+\alpha\right)=\frac{109}{50}$ and
$\bar{p}=n\beta=\frac{238}{25}$.

###  3.2. Improving Theorem 1.1 in [WZZ23] and addressing an open question in
Remark 1.2 of [WZZ23]

Herein we want to compare Theorem 2.4 and [WZZ23, Theorem 1.1], both dealing
with blow-up solutions to attraction-repulsion models with logistics. First,
in order to have consistency between these results, we have to fix $m_{2}=1$
in ($\mathcal{H}_{4}$). Additionally, for the ease of the reader, let us also
rephrase the related blow-up assumptions:

(17) $\text{Blow-up conditions in Theorem
\ref{ThBlowUp}:}\qquad\alpha>\beta\quad\text{and}\quad\begin{cases}\alpha>\max\bigl{\\{}m_{1}+\frac{2}{n}k-1,k-1\bigr{\\}}&\text{
if }m_{1}\geq 0,\\\ \alpha>\max\bigl{\\{}\frac{2}{n}k-1,k-1\bigr{\\}}&\text{
if }m_{1}<0,\\\ \end{cases}$

and

(18) $\text{Blow-up conditions in \cite[cite]{[\@@bibref{}{wang2023blow}{}{},
Theorem
1.1]}:}\qquad\alpha>\beta\quad\text{and}\quad\begin{cases}\alpha>\max\bigl{\\{}m_{1}+\frac{2}{n}k-1,k-1\bigr{\\}}&\text{
if }m_{1}>1,\\\ \alpha>\max\bigl{\\{}\frac{2}{n}k,k-1\bigr{\\}}&\text{ if
}m_{1}\leq 1.\\\ \end{cases}$

We easily note that if $m_{1}\geq 1$ the two conditions (17) and (18)
coincide. Now, we analyze the cases $m_{1}\leq 0$ and $0<m_{1}<1$, separately.

  * ➢

Case $m_{1}\leq 0$. By comparing
$\alpha>\max\bigl{\\{}\frac{2}{n}k-1,k-1\bigr{\\}}$ in (17) and
$\alpha>\max\bigl{\\{}\frac{2}{n}k,k-1\bigr{\\}}$ in (18), we observe that
(17) provides a larger range of values of $\alpha$ for which blow-up occurs
than (18) does whenever $n\in\\{1,2\\}$ or $n\geq 3$ provided
$1<k<\frac{n}{n-2}$.

  * ➢

Case $0<m_{1}<1$. The conditions above become
$\alpha>\max\bigl{\\{}m_{1}+\frac{2}{n}k-1,k-1\bigr{\\}}$ and
$\alpha>\max\bigl{\\{}\frac{2}{n}k,k-1\bigr{\\}}$, respectively. In
particular, thanks to the fact that $m_{1}<1$, also in this situation a
sharper condition is achieved for $n\in\\{1,2\\}$ or $n\geq 3$, under the
assumptions $1<k<\frac{n}{n-2}$ and $0<m_{1}<\frac{k(n-2)}{n}$ or
$1<k<\frac{n}{n-2}$ and $\frac{k(n-2)}{n}<m_{1}<1$.

From the above analysis, it is seen that Theorem 2.4 improves [WZZ23, Theorem
1.1]; additionally, it establishes that [WZZ23, (1.15), Theorem 1.1] is not
optimal, so giving an answer to an open question left in [WZZ23, Remark 1.2].

###  3.3. On the automatic applicability of Theorems 2.2 and 2.3 in some
related chemotaxis contexts

We can observe what follows:

  * ➢

Once the blow-up constrains in ($\mathcal{H}_{4}$) are accomplished, and
taking into account $m_{2}=m_{3}>0$, assumptions ($\mathcal{H}_{1}$) and
($\mathcal{H}_{2}$) are immediately satisfied; subsequently, Theorems 2.2 and
2.3 are applicable to unbounded solutions to model (8).

  * ➢

Theorems 2.2 and 2.3 can also be used in models close to (8), for which
unbounded solutions can be detected (see $\S$1); in particular, for
attraction-repulsion (linear and nonlinear) models with or without logistic
and general production laws ([WZZ23, LL21]), or for only attraction ones
([WL21, BFL21, Win18, Tan22, YMXD21]).

###  3.4. An open problem

As far as we know, establishing conditions ensuring blow-up solutions for the
case $m_{2}\neq m_{3}$ in model (8), is still an open problem; in particular,
we will give some details on related technical difficulties connected to this
issue in Remark 4.

##  4\. Local existence and necessary parameters

By an adaption of standard reasoning in the frame of the fixed-point theorem,
we can show the following result on local existence and extensibility of
classical solutions to (8).

######  Lemma 4.1.

Let $\Omega$ be a bounded and smooth domain of $\mathbb{R}^{n}$, with $n\geq
1$, $\chi,\xi,\lambda,\mu>0$, $m_{1},m_{2},m_{3}\in\mathbb{R}$, $k>1$, and let
$f_{i}$ and $u_{0}$ comply with (10). Then there exist $T_{max}\in(0,\infty]$
and a unique solution $(u,v,w)$ to problem (8), defined in
$\Omega\times(0,T_{max})$ and such that

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

Additionally, one has $u\geq 0$ in $\Omega\times(0,T_{max})$,

(19) $\int_{\Omega}u\leq M:=\max\\{M_{0},C\\}\quad\text{for all
}t\in[0,T_{max}),$

where $M_{0}=\displaystyle\int_{\Omega}u_{0}(x)\,dx$ and
$C:=\left(\frac{\lambda}{\mu}|\Omega|^{k-1}\right)^{\frac{1}{k-1}}$, and

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

Furthermore, if $u_{0}$ satisfies also the symmetrical assumptions in (12) and
$\Omega=B_{R}(0)$, with some $R>0$, then $u,v$ and $w$ are radially symmetric
with respect to $|x|$ in $\Omega\times(0,T_{max})$.

###### Proof.

The proof can be achieved by following well established results: for instance,
we refer the interested reader to [CW08, Nag95, Wan16, WD10]. In particular,
an integration of the first equation in (8) and an application of the Hölder
inequality give bound (19). ∎

Let us start with the following technical

######  Lemma 4.2.

Let $n\in\mathbb{N}$, $m_{1},m_{2},m_{3},\alpha,\beta$ be as in Assumptions
2.1. Moreover, for $\mathfrak{p}>\frac{n}{2}(m_{2}-m_{1}+\alpha)$ let

$\bar{p}>\max\begin{Bmatrix}\mathfrak{p}\vspace{0.1cm}\\\
1-m_{1}(n+2)\vspace{0.1cm}\\\ 1-m_{2}\vspace{0.1cm}\\\
1-m_{3}\vspace{0.1cm}\\\ 2-m_{1}-\frac{2}{n}\vspace{0.1cm}\\\
n\alpha\vspace{0.1cm}\\\ n\beta\vspace{0.1cm}\\\
m_{2}(n+2)(n+1)\vspace{0.1cm}\\\ m_{3}(n+2)(n+1)\end{Bmatrix},$
$\sigma\coloneqq\frac{2(p+m_{2}+\alpha-1)}{p+m_{1}-1},\quad\hat{\sigma}\coloneqq\frac{2p}{p+m_{1}-1},\quad\gamma\coloneqq\frac{\frac{m_{1}-m_{2}-\alpha}{2}+\frac{p}{n}+\frac{m_{2}+\alpha-1}{n}}{\frac{m_{1}-m_{2}-\alpha}{2}+\frac{p}{n}},\quad\delta\coloneqq\frac{p+m_{2}+\alpha-1}{p}.$

Then for all $p\geq\bar{p}$ these relations hold

(20a) $p>\frac{n}{2}(1-m_{1}),$ |  (20b) $0<\theta\coloneqq\frac{\frac{p+m_{1}-1}{2\mathfrak{p}}-\frac{p+m_{1}-1}{2(p+m_{2}+\alpha-1)}}{\frac{p+m_{1}-1}{2\mathfrak{p}}+\frac{1}{n}-\frac{1}{2}}<1,$
---|---
(20c) $0<\frac{\sigma\theta}{2}<1,$ |  (20d) $0<\hat{\theta}=\frac{\frac{p+m_{1}-1}{2}-\frac{p+m_{1}-1}{2p}}{\frac{p+m_{1}-1}{2}+\frac{1}{n}-\frac{1}{2}}<1,$
(20e) $0<\frac{\hat{\sigma}\hat{\theta}}{2}<1,$ |  (20f) $0<\frac{p+m_{3}-1}{p+m_{2}+\alpha-1}<1,$
(20g) $0<\frac{\beta}{p+m_{2}+\alpha-1}<1,$ |  (20h) $\frac{p+m_{3}-1}{p+m_{2}+\alpha-1}+\frac{\beta}{p+m_{2}+\alpha-1}<1,$
(20i) $0<\frac{p}{p+m_{2}+\alpha-1}<1,$ |  (20j) $\gamma>\delta>1,$
(20k) $0<\bar{\theta}\coloneqq\frac{\frac{p+m_{1}-1}{2p}-\frac{p+m_{1}-1}{2(p+m_{2}+\alpha-1)}}{\frac{p+m_{1}-1}{2p}+\frac{1}{n}-\frac{1}{2}}<1,$ |  (20l) $0<\frac{\sigma\bar{\theta}}{2}<1,$ (20m) $0<\frac{p+m_{3}-1}{p}<1.$

###### Proof.

To show our relations, we will need $p>1-m_{1}$ and $p>1+\beta-m_{2}-\alpha$.
As to the first inequality, due to ($\mathcal{H}_{1}$) and the restriction on
$\mathfrak{p}$, we have that $p\geq\bar{p}>\mathfrak{p}>1\geq 1-m_{1}$ for
$m_{1}\geq 0$, whilst for $m_{1}<0$ it suddenly derives from
$p\geq\bar{p}>1-m_{1}(n+2)$. For the second lower bound, assumptions
($\mathcal{H}_{2}$) or ($\mathcal{H}_{2}$), together with the definition of
$\bar{p}$, give $p\geq\bar{p}>1-m_{3}>1+\beta-m_{2}-\alpha$. Besides,
$m_{2}+\alpha>1$ is automatically true through ($\mathcal{H}_{2}$), or
alternatively by means of assumption ($\mathcal{H}_{1}$) in conjunction with
one among ($\mathcal{H}_{2}$), ($\mathcal{H}_{3}$) or ($\mathcal{H}_{3}$). The
same condition $m_{2}+\alpha>1$, $p\geq\bar{p}$ and the restriction on
$\mathfrak{p}$ and $p>1-m_{1}$ ensure relations (20a), (20b), (20c), (20k) and
(20l). Moreover, from (20a), $p>1-m_{1}$ and $p>2-m_{1}-\frac{2}{n}$, we
obtain (20d) and by using also $m_{1}>1-\frac{2}{n}$, we get (20e). On the
other hand, restrictions (20f), (20g), (20h), (20i) and (20m) come from the
definition of $\bar{p}$ in conjunction with $m_{2}+\alpha>m_{3}$, $p>1+\beta-
m_{2}-\alpha$, $m_{2}+\alpha>m_{3}+\beta$ and $m_{2}+\alpha>1$, respectively.
Finally, from ($\mathcal{H}_{1}$) it follows that $m_{2}+\alpha>m_{1}$, which
combined with $m_{2}+\alpha>1$ gives (20j). ∎

######  Remark 2.

For reasons which will be exploited later on, and precisely in Lemma 5.1, it
appears important to point out that assumptions ($\mathcal{H}_{2}$),
($\mathcal{H}_{3}$) and ($\mathcal{H}_{3}$) imply that the ratios (20f), (20h)
and (20m) in Lemma 4.2 can be also taken equal to 1.

##  5\. A priori estimates and proof of Theorems 2.2 and 2.3

In this section we will use ($\mathcal{H}_{1}$)— ($\mathcal{H}_{3}$).
Moreover, without explicitly computing their values, we underline that the
constants $c_{i}$ appearing below and throughout the paper depend inter alia
on $p$, are positive and their subscripts $i$ start anew in each new proof.

######  Lemma 5.1.

Under the hypotheses of Lemma 4.2, let $p=\bar{p}$ and $\mathfrak{p}$ be any
of the constants therein defined. If $(u,v,w)$ is a classical solution to
problem (8), $u\in L^{\infty}((0,T_{max});L^{\mathfrak{p}}(\Omega))$ and
$\varphi(t)$ is the energy function

$\varphi(t)\coloneqq\frac{1}{p}\int_{\Omega}(u+1)^{p}\quad\text{on
$(0,T_{max})$},$

then there exist $c\textsubscript{1},c\textsubscript{2}$ such that

(21)
$\varphi^{\prime}(t)\leq-c\textsubscript{1}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{2}\quad\text{for
all $t\in(0,T_{max})$}.$

###### Proof.

Let us differentiate the functional
$\varphi(t)=\frac{1}{p}\int_{\Omega}(u+1)^{p}$. Using the first equation of
(8) and the divergence theorem we have for every $t\in(0,T_{max})$

$\begin{split}\varphi^{\prime}(t)=\int_{\Omega}(u+1)^{p-1}u_{t}=&\int_{\Omega}(u+1)^{p-1}\nabla\cdot((u+1)^{m_{1}-1}\nabla
u)-\chi\int_{\Omega}(u+1)^{p-1}\nabla\cdot(u(u+1)^{m_{2}-1}\nabla v)\\\
&+\xi\int_{\Omega}(u+1)^{p-1}\nabla\cdot(u(u+1)^{m_{3}-1}\nabla
w)+\lambda\int_{\Omega}(u+1)^{p-1}u-\mu\int_{\Omega}(u+1)^{p-1}u^{k}\\\
=&-(p-1)\int_{\Omega}(u+1)^{p+m_{1}-3}\lvert\nabla
u\rvert^{2}+(p-1)\chi\int_{\Omega}u(u+1)^{p+m_{2}-3}\nabla u\cdot\nabla v\\\
&-(p-1)\xi\int_{\Omega}u(u+1)^{p+m_{3}-3}\nabla u\cdot\nabla
w+\lambda\int_{\Omega}(u+1)^{p-1}u-\mu\int_{\Omega}(u+1)^{p-1}u^{k}.\end{split}$

For $j\in\\{m_{2},m_{3}\\}$, we now define

$F_{j}(u)=\int_{0}^{u}\hat{u}(\hat{u}+1)^{p+j-3}d\hat{u},$

so observing that

(22) $0\leq F_{j}(u)\leq\frac{1}{p+j-1}\left[(u+1)^{p+j-1}-1\right].$

By considering the definition of $F_{j}(u)$ above, again the divergence
theorem, the second and third equation of (8), we have for every
$t\in(0,T_{max})$,

(23)
$\begin{split}\varphi^{\prime}(t)\leq&-c\textsubscript{3}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{4}\int_{\Omega}\nabla
F_{m_{2}}(u)\cdot\nabla v-c\textsubscript{5}\int_{\Omega}\nabla
F_{m_{3}}(u)\cdot\nabla
w+c\textsubscript{6}\int_{\Omega}(u+1)^{p}-c\textsubscript{7}\int_{\Omega}(u+1)^{p-1}u^{k}\\\
=&-c\textsubscript{3}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}-c\textsubscript{4}\int_{\Omega}F_{m_{2}}(u)\Delta
v+c\textsubscript{5}\int_{\Omega}F_{m_{3}}(u)\Delta
w+c\textsubscript{6}\int_{\Omega}(u+1)^{p}-c\textsubscript{7}\int_{\Omega}(u+1)^{p-1}u^{k}.\end{split}$

Let us now specify how each of the constrains in Assumptions 2.1 takes part in
our computation.

First, from ($\mathcal{H}_{1}$) we have $\mathfrak{p}>1$; this makes
meaningful our assumption $u\in
L^{\infty}((0,T_{max});L^{\mathfrak{p}}(\Omega))$. (Recall that $u\in
L^{\infty}((0,T_{max});L^{1}(\Omega))$ is always met by (19).)

Now, by exploiting ($\mathcal{H}_{2}$) and (22), we can see that from (23), if
we neglect the nonpositive terms we get on $(0,T_{max})$

(24)
$\varphi^{\prime}(t)\leq-c\textsubscript{3}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{8}\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}+c\textsubscript{9}\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{\beta}+c\textsubscript{6}\int_{\Omega}(u+1)^{p}.$

As to the third term, by using twice Hölder’s inequality (recall (20f) and
(20g)), we obtain, for every $t\in(0,T_{max})$,

(25)
$c\textsubscript{9}\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{\beta}\leq
c\textsubscript{10}\left(\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}\right)^{\frac{p+m_{3}-1}{p+m_{2}+\alpha-1}}\left(\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}\right)^{\frac{\beta}{p+m_{2}+\alpha-1}}.$

Moreover, since for any $\varepsilon>0$ there is $d(\varepsilon)>0$ such that
this inequality (see [FV21, Lemma 4.3])

$A^{d_{1}}B^{d_{2}}\leq\varepsilon(A+B)+d(\varepsilon),\quad A,B\geq
0,\,d_{1},d_{2}>0,\,d_{1}+d_{2}<1,$

is true, by virtue of (20h), we have that

(26)
$c\textsubscript{9}\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{\beta}\leq\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}+c\textsubscript{11}\quad\textrm{for
all $t\in(0,T_{max})$}.$

Through (20i), an application of Young’s inequality provides

(27) $c\textsubscript{6}\int_{\Omega}(u+1)^{p}\leq
c\textsubscript{12}\left(\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}\right)^{\frac{p}{p+m_{2}+\alpha-1}}\leq\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}+c\textsubscript{13}\quad\textrm{on
}(0,T_{max}).$

To control the term $\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}$, we invoke the
Gagliardo–Nirenberg and Young’s inequalities, so to bound the mentioned
integral with $\int_{\Omega}|\nabla(u+1)^{\frac{p+m_{1}-1}{2}}|^{2}$. More
exactly by relying on relations (20b) and (20c), boundedness of
$\int_{\Omega}(u+1)^{\mathfrak{p}}$ and of $\int_{\Omega}u$ provide for any
$L_{1}>0$

(28)
$\begin{split}L_{1}\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}=&L_{1}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\frac{2(p+m_{2}+\alpha-1)}{p+m_{1}-1}}_{L^{\frac{2(p+m_{2}+\alpha-1)}{p+m_{1}-1}}(\Omega)}\\\
\leq&c\textsubscript{14}\left(\left\lVert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\sigma\theta}_{L^{2}(\Omega)}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\sigma(1-\theta)}_{L^{\frac{2\mathfrak{p}}{p+m_{1}-1}}(\Omega)}+\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}^{\sigma}\right)\\\
\leq&c\textsubscript{15}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\right)^{\frac{\sigma\theta}{2}}+c\textsubscript{16}\leq\frac{c_{3}}{2}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{17}\quad\textrm{for
all }t\in(0,T_{max}).\end{split}$

(We underline that herein we have used the elementary inequality

(29) $C_{1}(\tau)\left(A^{\tau}+B^{\tau}\right)\leq(A+B)^{\tau}\leq
C_{2}(\tau)(A^{\tau}+B^{\tau})\quad\textrm{for all }A,B\geq 0,\tau>0\textrm{
and proper }C_{1},C_{2}>0,$

which might tacitly be used in the next lines.) Putting together (24), (26),
(27), (28), we have the claim.

If assumption ($\mathcal{H}_{2}$) is complied, bound (27) can be replaced for
any $L_{2}>0$ with

(30)
$\begin{split}L_{2}\int_{\Omega}(u+1)^{p}=&L_{2}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\frac{2p}{p+m_{1}-1}}_{L^{\frac{2p}{p+m_{1}-1}}(\Omega)}\\\
\leq&c\textsubscript{18}\left(\left\lVert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\hat{\sigma}\hat{\theta}}_{L^{2}(\Omega)}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\hat{\sigma}(1-\hat{\theta)}}_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}+\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2}{p+m_{1}-1}}(\Omega)}^{\hat{\sigma}}\right)\\\
\leq&c\textsubscript{18}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\right)^{\frac{\hat{\sigma}\hat{\theta}}{2}}+c\textsubscript{19}\leq\frac{c_{3}}{4}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{20}\quad\textrm{on
}(0,T_{max}),\end{split}$

where in this last step we used Young’s inequality in conjunction with (20d)
and (20e). Moreover, by taking $m_{2}+\alpha=m_{3}+\beta$ in
($\mathcal{H}_{2}$) (recall Remark 2), in estimate (25) the powers at the rhs
have 1 as sum; subsequently, up to constants, it becomes (26). So, by
considering (24), (26), (28) and (30) we have the claim when either
($\mathcal{H}_{1}$) and ($\mathcal{H}_{2}$) or ($\mathcal{H}_{1}$) and
($\mathcal{H}_{2}$) are used.

Let us show how achieving the same conclusion by applying either
($\mathcal{H}_{1}$) and ($\mathcal{H}_{3}$) or ($\mathcal{H}_{1}$) and
($\mathcal{H}_{3}$). The main idea is alternatively treating in relation (24)
the term

$\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{\beta}\quad\textrm{on
}(0,T_{max}).$

More specifically, if $\beta\in\left(0,1\right]$ and we take into account the
boundedness of $\int_{\Omega}u$ on $(0,T_{max})$, we have that

(31) $\int_{\Omega}(u+1)^{p+m_{3}-1}\int_{\Omega}(u+1)^{\beta}\leq
c\textsubscript{21}\int_{\Omega}(u+1)^{p+m_{3}-1}\quad\textrm{for all
}t\in(0,T_{max}).$

In turn, by relying respectively on assumption ($\mathcal{H}_{3}$) or
($\mathcal{H}_{3}$) (both with strict inequality), by means of Young’s
inequality, thanks to (20m) and (20f), it is also possible to see that

(32)
$c\textsubscript{21}\int_{\Omega}(u+1)^{p+m_{3}-1}\leq\int_{\Omega}(u+1)^{p}+c\textsubscript{22}\quad\textrm{on
}(0,T_{max}),$

or

(33)
$c\textsubscript{21}\int_{\Omega}(u+1)^{p+m_{3}-1}\leq\int_{\Omega}(u+1)^{p+m_{2}-1+\alpha}+c\textsubscript{23}\quad\textrm{for
all }t\in(0,T_{max}).$

As before, bounds (24), (31), (32), or alternatively (33), (30) and (28), lead
to the same conclusion. (For the limit cases in ($\mathcal{H}_{3}$) or
($\mathcal{H}_{3}$), namely $m_{3}=1$ or $m_{2}+\alpha=m_{3}$, by relying
again on Remark 2, we can directly exploit estimate (31), without using
Young’s inequality, and conclude as above.) ∎

The next step consists in ensuring some time independent estimate of $u$ in
the $L^{\bar{p}}(\Omega)$-norm.

######  Lemma 5.2.

Let Lemma 5.1 be true. Then $u\in
L^{\infty}((0,T_{max});L^{\bar{p}}(\Omega))$.

###### Proof.

With a view to inequality (21), as already done in (30), a further application
of the Gagliardo–Nirenberg inequality, supported by (20d), leads also thanks
to (29) to

$\begin{split}\int_{\Omega}(u+1)^{p}\leq
c\textsubscript{1}\left(\left\lVert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\hat{\sigma}\hat{\theta}}_{L^{2}(\Omega)}+1\right)\leq
c\textsubscript{2}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+1\right)^{\frac{{\hat{\sigma}\hat{\theta}}}{2}}\quad\textrm{on
}(0,T_{max}),\end{split}$

or, equivalently

(34)
$-c\textsubscript{3}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\leq-\left(\int_{\Omega}(u+1)^{p}\right)^{\frac{2}{\hat{\sigma}\hat{\theta}}}+c\textsubscript{3}\quad\text{for
every $t\in(0,T_{max})$}.$

Finally, by using relations (21) and (34), we arrive at this initial problem

$\begin{dcases}\varphi^{\prime}(t)\leq
c\textsubscript{4}-c\textsubscript{5}\varphi(t)^{\frac{2}{\hat{\sigma}\hat{\theta}}}\quad\textrm{for
every }t\in(0,T_{max}),\\\
\varphi(0)=\frac{1}{p}\int_{\Omega}(u_{0}+1)^{p},\end{dcases}$

which ensures that
$\int_{\Omega}u^{\bar{p}}\int_{\Omega}u^{p}\leq\int_{\Omega}(u+1)^{p}\leq
p\max\left\\{\varphi(0),\left(\frac{c\textsubscript{4}}{c\textsubscript{5}}\right)^{\frac{\hat{\sigma}\hat{\theta}}{2}}\right\\}$
for all $t<T_{max}$. ∎

By taking advantage from the previous lemma, let us show the uniform-in-time
boundedness of $u$.

######  Lemma 5.3.

Under the hypotheses of Lemma 5.2, $u\in
L^{\infty}((0,T_{max});L^{\infty}(\Omega))$.

###### Proof.

With the same nomenclature used by Tao and Winkler, $u$ also classically
solves in $\Omega\times(0,T_{max})$ problem (A.1) of [TW12, Appendix A] for

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

In particular, also taking into account the boundary condition on $v$ and $w$,
we can see that (A.2)–(A.5) are met. On the other hand, for any
$\lambda,\mu>0$ and $k>1$, it holds that $\lambda u-\mu u^{k}$ has a positive
maximum $L$ at $u_{M}=\left(\frac{\lambda}{k\mu}\right)^{\frac{1}{k-1}}$, so
that from $g(x,t)\leq L$ in $\Omega\times(0,T_{max})$ the second inclusion of
(A.6) is accomplished for any choice of $q_{2}$. As to (A.7)–(A.10), let us
first define the quantities

$l_{1}(q_{1})=1-m_{1}\frac{(n+1)q_{1}-(n+2)}{q_{1}-(n+2)},\quad
l_{2}(q_{2})=1-m_{1}\frac{1}{1-\frac{nq_{2}}{(n+2)(q_{2}-1)}}\quad\textrm{and}\quad
l_{3}=\frac{n}{2}\left(1-m_{1}\right).$

Recalling the definition and properties of $p=\bar{p}$, we have
$p>1-m_{1}(n+2)$ and henceforth for any $m_{1}\in\mathbb{R}$, it holds
$1-m_{1}(n+2)\geq l_{1}((n+2)(n+1))$ and there exists $\tilde{q}_{2}$ large
enough so to have $1-m_{1}(n+2)\geq l_{2}(\tilde{q}_{2})$. Subsequently,
[TW12, (A.7)] (i.e. $u\in L^{\infty}((0,T_{max});L^{p}(\Omega))$, that is
$\int_{\Omega}u^{p}\leq c\textsubscript{1}$) is obviously accomplished, [TW12,
(A.8), Lemma A.1.] is fulfilled for $q_{1}=(n+2)(n+1)$, [TW12, (A.9), Lemma
A.1.] for sufficiently large $q_{2}$, whereas [TW12, (A.10), Lemma A.1.] is
directly true thanks to (20a). Let us dedicate to the first inclusion of
[TW12, (A.6)]. Starting from the gained bound of $u$, let us exploit elliptic
regularity results applied to the second and third equation of system (8); in
particular, we only analyze $-\Delta
v=f_{1}(u)-\frac{1}{|\Omega|}\int_{\Omega}f_{1}(u)$, being the case for $w$
equivalent. Let us observe that from relation (11) on $f_{1}$, and
$p=\bar{p}>n\alpha$ naturally implying $(1+u)^{\alpha}\leq(1+u)^{p}$, we have
for all $t\in(0,T_{max})$

$\int_{\Omega}|f_{1}(u)-\frac{1}{|\Omega|}\int_{\Omega}f_{1}(u)|^{\frac{p}{\alpha}}\leq
c\textsubscript{2}\int_{\Omega}(1+u)^{p}+c\textsubscript{3}\int_{\Omega}\left(\int_{\Omega}(1+u)^{\alpha}\right)^{\frac{p}{\alpha}}\leq
c\textsubscript{2}\int_{\Omega}(1+u)^{p}+c\textsubscript{3}\int_{\Omega}\left(\int_{\Omega}(1+u)^{p}\right)^{\frac{p}{\alpha}}\leq
c\textsubscript{4},$

which gives $f_{1}(u)-\frac{1}{|\Omega|}\int_{\Omega}f_{1}(u)\in
L^{\infty}((0,T_{max});L^{\frac{p}{\alpha}}(\Omega))$, in turn $v\in
L^{\infty}((0,T_{max});W^{2,\frac{p}{\alpha}}(\Omega))$ and, hence, through
the Sobolev embeddings $\nabla v\in
L^{\infty}((0,T_{max});W^{1,\frac{p}{\alpha}}(\Omega))\xhookrightarrow{}L^{\infty}((0,T_{max});L^{\infty}(\Omega))$.
Consequently, thanks to the Hölder inequality (recall that from Lemma 4.2 one
has that $p>m_{2}(n+2)(n+1)$), by using the uniform-in-time boundedness of $u$
in $L^{\bar{p}}(\Omega)$ we can write on $(0,T_{max})$

$\begin{split}\int_{\Omega}|u(u+1)^{m_{2}-1}\nabla
v|^{(n+2)(n+1)}&\leq\lVert\nabla
v(\cdot,t)\rVert_{L^{\infty}(\Omega)}^{(n+2)(n+1)}|\Omega|^{\frac{p-m_{2}(n+2)(n+1)}{p}}\left(\int_{\Omega}(u+1)^{p}\right)^{\frac{m_{2}(n+2)(n+1)}{p}}\leq
c\textsubscript{5}.\end{split}$

Reasoning in a similar way on the third equation of (8), we have $\nabla w\in
L^{\infty}((0,T_{max});L^{\infty}(\Omega))$,

$\begin{split}\int_{\Omega}|u(u+1)^{m_{3}-1}\nabla w|^{(n+2)(n+1)}\leq
c\textsubscript{6}\quad\textrm{for all }t\in(0,T_{max}),\end{split}$

and as a consequence

$f=\chi u(u+1)^{m_{2}-1}\nabla v-\xi u(u+1)^{m_{3}-1}\nabla w\in
L^{\infty}((0,T_{max});L^{(n+2)(n+1)}(\Omega)).$

Since all the hypotheses of [TW12, Lemma A.1.] are fulfilled, we have the
claim. ∎

Now we are in a position to prove our first results.

#### Proof of Theorem 2.2:

Let $(u,v,w)$ be a given blow-up solution at some finite time $T_{max}$ to
problem (8). If $u$ was not unbounded in some $L^{\mathfrak{p}}(\Omega)$-norm,
Lemma 5.3 would imply the uniform-in-time boundedness of $u$, contradicting
hypothesis (13). ∎

#### Proof of Theorem 2.3:

By making use of (26) only or altogether (31), (32) and (33), we observe that
bound (24) can essentially be reorganized as

(35)
$\varphi^{\prime}(t)\leq-c\textsubscript{1}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{2}\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}+c\textsubscript{3}\int_{\Omega}(u+1)^{p}+c\textsubscript{4}\quad\textrm{on
}(0,T_{max}).$

From the one hand, as already done in (27), up to a constant the Young
inequality yields

(36)
$\int_{\Omega}(u+1)^{p}\leq\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}+c\textsubscript{5}\quad\textrm{for
all }t\in(0,T_{max});$

moreover, by recalling (20k) and (20l), we can derive again through the
Gagliardo–Nirenberg and Young’s inequalities, this bound valid for any $K>0$

(37)
$\begin{split}K\int_{\Omega}(u+1)^{p+m_{2}+\alpha-1}=&K\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\frac{2(p+m_{2}+\alpha-1)}{p+m_{1}-1}}_{L^{\frac{2(p+m_{2}+\alpha-1)}{p+m_{1}-1}}(\Omega)}\\\
\leq&c\textsubscript{6}\left(\left\lVert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\sigma\bar{\theta}}_{L^{2}(\Omega)}\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert^{\sigma(1-\bar{\theta})}_{L^{\frac{2p}{p+m_{1}-1}}(\Omega)}+\left\lVert(u+1)^{\frac{p+m_{1}-1}{2}}\right\rVert_{L^{\frac{2p}{p+m_{1}-1}}(\Omega)}^{\sigma}\right)\\\
\leq&c\textsubscript{7}\left(\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}\right)^{\frac{\sigma\bar{\theta}}{2}}\left(\int_{\Omega}(u+1)^{p}\right)^{\frac{p+m_{1}-1}{2p}\sigma(1-\bar{\theta})}+c\textsubscript{8}\left(\int_{\Omega}(u+1)^{p}\right)^{\frac{p+m_{1}-1}{2p}\sigma}\\\
\leq&c\textsubscript{1}\int_{\Omega}\left\lvert\nabla(u+1)^{\frac{p+m_{1}-1}{2}}\right\rvert^{2}+c\textsubscript{9}\left(\int_{\Omega}(u+1)^{p}\right)^{\gamma}+c\textsubscript{10}\left(\int_{\Omega}(u+1)^{p}\right)^{\delta}\quad\textrm{on
}(0,T_{max}).\end{split}$

Finally, by inserting relations (36) and (37) into (35), and by using (37)
with $K=c\textsubscript{2}+c\textsubscript{3}$, we obtain for proper positive
$\mathcal{A},\mathcal{B}$ and $\mathcal{C}$

(38)
$\varphi^{\prime}(t)\leq\mathcal{A}\varphi^{\gamma}(t)+\mathcal{B}\varphi^{\delta}(t)+\mathcal{C}\quad\textrm{on
}(0,T_{max}).$

Since $p=\bar{p}>\mathfrak{p}$, from Theorem 2.2 we know that
$\limsup_{t\rightarrow T_{max}}\frac{1}{p}\int_{\Omega}(u+1)^{p}=\infty.$ On
the other hand, since $\varphi(t)$ satisfies relation (38) for any
$0<t<T_{max}$, the function
$\Psi(\xi)=\mathcal{A}\xi^{\gamma}+\mathcal{B}\xi^{\delta}+\mathcal{C}$ obeys
the Osgood criterion (14). Thereafter, by integrating (38) between $0$ and
$T_{max}$, we obtain estimate (15), and the first conclusion is achieved.

As to the derivation of the explicit expression for the lower bound $T$, let
us reduce (38) as follows: from
$|\Omega|\leq\int_{\Omega}(u+1)^{p}=p\varphi(t)$, we can estimate
$\mathcal{C}$ in relation (38) as

$\mathcal{C}\leq
p\frac{\mathcal{C}}{|\Omega|}\varphi(t)=:\bar{\mathcal{C}}\varphi(t),$

so that (38) can be rewritten in this form:

(39)
$\varphi^{\prime}(t)\leq\mathcal{A}\varphi^{\gamma}(t)+\mathcal{B}\varphi^{\delta}(t)+\bar{\mathcal{C}}\varphi(t)\quad\textrm{on
}(0,T_{max}).$

Now, since $\varphi$ blows up at finite time $T_{max}$, there exists a time
$t_{1}\in[0,T_{max})$ such that

$\varphi(t)\geq\varphi(0)\quad\textrm{for all }\,t\geq t_{1}\in[0,T_{max}).$

From $\gamma>\delta>1$ (recall (20j)), we can estimate the second and third
term on the rhs of (39) by means of $\varphi^{\gamma}$:

(40)
$\varphi^{\delta}(t)\leq\varphi(0)^{\delta-\gamma}\varphi^{\gamma}(t)\quad\textrm{
and
}\quad\varphi(t)\leq\varphi(0)^{1-\gamma}\varphi^{\gamma}(t)\quad\textrm{for
all }\,t\geq t_{1}\in[0,T_{max}).$

By plugging expressions (40) into (39) we obtain for

$\mathcal{D}=\mathcal{A}+\mathcal{B}\varphi(0)^{\delta-\gamma}+\bar{\mathcal{C}}\varphi(0)^{1-\gamma},$
(41) $\varphi^{\prime}(t)\leq\mathcal{D}\varphi^{\gamma}(t)\quad\textrm{for
all }\,t\geq t_{1}\in[0,T_{max}),$

so that an integration of (41) on $(t_{1},T_{max})$ yields this explicit lower
bound for $T_{max}$:

$\displaystyle
T=\frac{\varphi(0)^{1-\gamma}}{\mathcal{D}(\gamma-1)}=\int_{\varphi(0)}^{\infty}\frac{d\tau}{\mathcal{D}\tau^{\gamma}}\leq\int_{t_{1}}^{T_{max}}d\tau\leq\int_{0}^{T_{max}}d\tau=T_{max}.$

∎

######  Remark 3.

For completeness, we observe that it is also possible to avoid estimate (36);
indeed, in relation (35) the term $\int_{\Omega}(u+1)^{p}$ is directly
$p\varphi(t)$, so that (38) would read

$\varphi^{\prime}(t)\leq\tilde{\mathcal{A}}\varphi^{\gamma}(t)+\tilde{\mathcal{B}}\varphi^{\delta}(t)+\tilde{\mathcal{C}}\varphi(t)+\tilde{\mathcal{D}}\quad\textrm{on
}(0,T_{max}).$

##  6\. Finite-time blow-up to a simplified version of problem (8)

This section is dedicated to prove finite time blow-up for solutions to
problem (8) in a more specific case; in our computations we will be inspired
by [Tan22, WZZ23], where respectively blow-up is established in a model with
only attraction, nonlinear diffusion and sensitivity and logistic term, and in
an attraction-repulsion one, with nonlinear diffusion but linear sensitivities
and logistics.

###  6.1. Detecting unbounded solutions to problem (8) for $m_{2}=m_{3}>0$

Let us fix $m_{2}=m_{3}>0$ in model (8), and in turn let us set $z=\chi v-\xi
w$, $m(t)=\chi m_{1}(t)-\xi m_{2}(t)$ and $f(u)=\chi f_{1}(u)-\xi f_{2}(u)$,
being $m_{1}(t)$ and $m_{2}(t)$ defined in (9); in this way, problem (8)
itself is reduced into

(42) $\begin{cases}u_{t}=\nabla\cdot((u+1)^{m_{1}-1}\nabla
u-u(u+1)^{m_{2}-1}\nabla z)+\lambda u-\mu u^{k}&\text{ in
}\Omega\times(0,T_{max}),\\\ 0=\Delta z-m(t)+f(u)&\text{ in
}\Omega\times(0,T_{max}),\\\ u_{\nu}=z_{\nu}=0&\text{ on
}\partial\Omega\times(0,T_{max}),\\\ u(x,0)=u_{0}(x)&x\in\bar{\Omega},\\\
\int_{\Omega}z(x,t)dx=0&\textrm{for all }t\in(0,T_{max}).\end{cases}$

In particular, if we confine our study to radially symmetric cases, by setting
$r:=|x|$ and by considering $\Omega=B_{R}(0)\subset\mathbb{R}^{n}$, $n\geq 1$
and some $R>0$, the radially symmetric local solution

$(u,z)=(u(r,t),z(r,t))$

to model (42) solves the following scalar problem

(43)
$\begin{cases}r^{n-1}u_{t}=(r^{n-1}(u+1)^{m_{1}-1}u_{r})_{r}-(r^{n-1}u(u+1)^{m_{2}-1}z_{r})_{r}+\lambda
r^{n-1}u-\mu r^{n-1}u^{k}&r\in(0,R),t\in(0,T_{max}),\\\
0=(r^{n-1}z_{r})_{r}-r^{n-1}m(t)+r^{n-1}f(u)&r\in(0,R),t\in(0,T_{max}),\\\
u_{r}=z_{r}=0&r=R,t\in(0,T_{max}),\\\ u(r,0)=u_{0}(r)&r\in(0,R),\\\
\int_{0}^{R}r^{n-1}z(r,t)\,dr=0&\textrm{for all }t\in(0,T_{max}).\end{cases}$

In the same spirit of [JL92], we introduce the mass accumulation function

(44)
$U(s,t):=\int_{0}^{s^{\frac{1}{n}}}\rho^{n-1}u(\rho,t)\,d\rho\quad\text{for
}s\in[0,R^{n}]\text{ and }t\in[0,T_{max}),$

which implies that

(45) $U_{s}(s,t)=\frac{1}{n}u(s^{\frac{1}{n}},t)\quad\text{and}\quad
U_{ss}(s,t)=\frac{1}{n^{2}}s^{\frac{1}{n}-1}u_{r}(s^{\frac{1}{n}},t)\quad\text{for
}s\in(0,R^{n})\text{ and }t\in(0,T_{max}).$

By the definition of $U$ and by exploiting (45), we obtain

(46)
$\begin{split}U_{t}(s,t)&=s^{1-\frac{1}{n}}(u(s^{\frac{1}{n}},t)+1)^{m_{1}-1}u_{r}(s^{\frac{1}{n}},t)-s^{1-\frac{1}{n}}u(s^{\frac{1}{n}},t)(u(s^{\frac{1}{n}},t)+1)^{m_{2}-1}z_{r}(s^{\frac{1}{n}},t)\\\
&+\lambda\int_{0}^{s^{\frac{1}{n}}}\rho^{n-1}u(\rho,t)\,d\rho-\mu\int_{0}^{s^{\frac{1}{n}}}\rho^{n-1}u^{k}(\rho,t)\,d\rho\\\
&=n^{2}s^{2-\frac{2}{n}}(nU_{s}(s,t)+1)^{m_{1}-1}U_{ss}(s,t)-ns^{1-\frac{1}{n}}U_{s}(s,t)(nU_{s}(s,t)+1)^{m_{2}-1}z_{r}(s^{\frac{1}{n}},t)\\\
&+\lambda U(s,t)-\mu
n^{k-1}\int_{0}^{s}U_{s}^{k}(\sigma,t)\,d\sigma\quad\text{for
}s\in(0,R^{n})\text{ and }t\in(0,T_{max}).\end{split}$

Besides, with an integration over $(0,r)$ of the second equation in (43) and
the substitution $r=s^{\frac{1}{n}}$, we arrive at

$z_{r}=\frac{m(t)}{n}s^{\frac{1}{n}}-\frac{1}{n}s^{\frac{1}{n}-1}\int_{0}^{s}f(nU_{s}(\sigma,t))\,d\sigma\quad\text{for
}s\in(0,R^{n})\text{ and }t\in(0,T_{max}),$

which inserted into relation (46) gives (observe $U\geq 0$)

(47) $\begin{split}U_{t}(s,t)&\geq
n^{2}s^{2-\frac{2}{n}}(nU_{s}+1)^{m_{1}-1}U_{ss}-sU_{s}(nU_{s}+1)^{m_{2}-1}m(t)+U_{s}(nU_{s}+1)^{m_{2}-1}\int_{0}^{s}f(nU_{s}(\sigma,t))\,d\sigma\\\
&\quad-\mu n^{k-1}\int_{0}^{s}U_{s}^{k}(\sigma,t)\,d\sigma\quad\text{for all
}s\in(0,R^{n})\text{ and }t\in(0,T_{max}).\end{split}$

In addition, given $s_{0}\in(0,R^{n})$, $\gamma\in(-\infty,1)$ and $U$ as in
(44), we introduce the moment-type functional

(48) $\phi(t):=\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)U(s,t)\,ds\quad\text{for
}t\in[0,T_{max}),$

which is well defined and belongs to $C^{0}([0,T_{max}))\cap
C^{1}((0,T_{max}))$. Moreover, we define

(49)
$\psi(t):=\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}^{m_{2}+\alpha}(s,t)\,ds\quad\text{for
}t\in(0,T_{max}),$

and the set

(50)
$S_{\phi}=\left\\{t\in(0,T_{max}):\phi(t)\geq\frac{M-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma}\right\\},$

where $M$ is the bound of the $L^{1}(\Omega)$-norm of $u$ established in (19)
and $\omega_{n}=n|B_{1}(0)|$. With these preparations in our hands, let us
give a series of necessary lemmas, some of which are not new.

We start with a result dealing with the concavity of $U$ and some estimate for
$m(t)$.

######  Lemma 6.1.

Let $f_{1},f_{2}$ and $u_{0}$ satisfy (10) and (12), $\alpha>\beta$ and
$\gamma\in\left(-\infty,1\right).$ Then the following relations hold:

$U_{ss}(s,t)\leq 0\quad\text{for all }s\in(0,R^{n})\text{ and
}t\in(0,T_{max}),$

and

(51) $m(t)\leq
c_{0}+\frac{1}{2s}\int_{0}^{s}f(nU_{s}(\sigma,t))\,d\sigma\quad\text{for
}s_{0}\in\left(0,\frac{R^{n}}{6}\right]\text{and for all }s\in(0,s_{0}),$

for all $t\in S_{\phi}$, and with some $c_{0}$.

###### Proof.

As to the concavity property, the proof is based on minor adjustments of
[WZZ23, Lemma 3.2] or [Win18, Lemma 2.2]. The remaining conclusion follows by
closely reasoning as in [LL21, Lemma 2.5]; we herein only add that
$\alpha>\beta$ is required just for the construction of

$c_{0}=\chi
f_{1}\left(\frac{8n}{2^{\gamma}(3-\gamma)\omega_{n}}\right)+\frac{1}{6}\left(\frac{\chi
k_{3}(\alpha-\beta)}{\beta}\left(\frac{2\xi k_{2}\beta}{\chi
k_{3}\alpha}\right)^{\frac{\alpha}{\alpha-\beta}}+L(\chi+2)\right)>0,$

being $L=L(K)$ estimated by

$|f(s)|,f_{1}(s),f_{2}(s)\leq L\quad\text{for }s\in[0,K],K>0.$

(We point out that $c_{0}$ will be used in some other places below.) ∎

Let us now start with the analysis of the temporal evolution of $\phi$.

######  Lemma 6.2.

Under the same assumptions of Lemma 6.1, let
$s_{0}\in\left(0,\frac{R^{n}}{6}\right]$. Then

(52) $\begin{split}\phi^{\prime}(t)&\geq
n^{2}\int_{0}^{s_{0}}s^{2-\frac{2}{n}-\gamma}(s_{0}-s)(nU_{s}+1)^{m_{1}-1}U_{ss}\,ds-
c_{0}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\,ds\\\
&\quad+\frac{1}{2}\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\left[\int_{0}^{s}f(nU_{s}(\sigma,t))\,d\sigma\right]\,ds\\\
&\quad-\mu
n^{k-1}\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)\left[\int_{0}^{s}U_{s}^{k}(\sigma,t)\,d\sigma\right]\,ds\\\
&=I_{1}+I_{2}+I_{3}+I_{4}\quad\text{for all }t\in S_{\phi}.\end{split}$

###### Proof.

By the definition of $\phi$ (recall (48)) and exploiting (47), we get this
estimate

$\begin{split}\phi^{\prime}(t)&=\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)U_{t}(s,t)\,ds\\\
&\geq
n^{2}\int_{0}^{s_{0}}s^{2-\frac{2}{n}-\gamma}(s_{0}-s)(nU_{s}+1)^{m_{1}-1}U_{ss}\,ds-\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}m(t)\,ds\\\
&\quad+\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\left[\int_{0}^{s}f(nU_{s}(\sigma,t))\,d\sigma\right]\,ds\\\
&\quad-\mu
n^{k-1}\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)\left[\int_{0}^{s}U_{s}^{k}(\sigma,t)\,d\sigma\right]\,ds\quad\text{for
all }t\in S_{\phi}.\end{split}$

Finally, by applying (51), we obtain the thesis. ∎

The next results provide some lower bounds of $I_{1},I_{2},I_{3}$ and $I_{4}$
in respect of $\psi(t)$ defined in (49).

######  Lemma 6.3.

Let $u_{0}$ satisfy the related assumptions in (10) and (12),
$m_{1}\in\mathbb{R}$, $m_{2},\alpha>0$, $k>1$ and $\gamma\in(-\infty,1)$.

  * ➢

If $m_{1},m_{2},\alpha$ and $\gamma$ comply with

$m_{2}+\alpha>m_{1}\quad\text{and}\quad\gamma>2-\frac{2}{n}\frac{(m_{2}+\alpha)}{(m_{2}+\alpha-
m_{1})}\quad\text{if}\quad m_{1}\geq 0,$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\gamma<2-\frac{2}{n}\quad\text{if}\quad
m_{1}<0,$

then there exist $\varepsilon>0$ (sufficiently small), and
$c\textsubscript{1}$, $c\textsubscript{2}$, $c\textsubscript{3}$,
$c\textsubscript{4}$, $c\textsubscript{5}$ such that for any
$s_{0}\in\left(0,\frac{R^{n}}{6}\right]$

$I_{1}\geq\begin{cases}-c\textsubscript{1}s_{0}^{(3-\gamma)\frac{(m_{2}+\alpha-
m_{1})}{m_{2}+\alpha}-\frac{2}{n}}\psi^{\frac{m_{1}}{m_{2}+\alpha}}(t)-c\textsubscript{2}s_{0}^{3-\gamma-\frac{2}{n}}&\text{if
}m_{1}>0,\\\
-c\textsubscript{3}s_{0}^{(3-\gamma)\frac{(m_{2}+\alpha-\varepsilon)}{m_{2}+\alpha}-\frac{2}{n}}\psi^{\frac{\varepsilon}{m_{2}+\alpha}}(t)-c\textsubscript{4}s_{0}^{3-\gamma-\frac{2}{n}}&\text{if
}m_{1}=0,\\\ -c\textsubscript{5}s_{0}^{3-\gamma-\frac{2}{n}}&\text{if
}m_{1}<0\end{cases}$

for all $t\in S_{\phi}$.

  * ➢

If $m_{2},\alpha,k$ and $\gamma$ are such that

$m_{2}+\alpha>k\quad\text{and}\quad 2-\frac{(m_{2}+\alpha)}{k}<\gamma<1,$

then there exists $c\textsubscript{6}$ such that for any
$s_{0}\in\left(0,\frac{R^{n}}{6}\right]$

$I_{4}\geq-c\textsubscript{6}\psi^{\frac{k}{m_{2}+\alpha}}(t)s_{0}^{(3-\gamma)\frac{(m_{2}+\alpha-k)}{m_{2}+\alpha}}\quad\text{for
all }t\in S_{\phi}.$

###### Proof.

The proof can be found in [Tan22, Lemmas 3.6 and 3.7]. ∎

As to the estimate of $I_{2}+I_{3}$ we need to rearrange some computations;
this is exactly where we have to go beyond the analysis in [Tan22] and
[WZZ23].

######  Lemma 6.4.

Under the same assumptions of Lemma 6.1, let moreover $m_{2},\alpha>0$ comply
with $m_{2}+\alpha>1.$ Then for some $c\textsubscript{1}$,
$c\textsubscript{2}$ and $c\textsubscript{3}$ we have

(53) $I_{2}+I_{3}\geq
c\textsubscript{1}\psi(t)-c\textsubscript{2}\psi^{\frac{m_{2}}{m_{2}+\alpha}}(t)s_{0}^{(3-\gamma)\frac{\alpha}{m_{2}+\alpha}}-c\textsubscript{3}s_{0}^{3-\gamma}\;\textrm{
for any }s_{0}\in\left(0,\frac{R^{n}}{6}\right]\textrm{ and for all }t\in
S_{\phi}.$

###### Proof.

Since $\alpha>\beta$, by applying the Young inequality and (12), we get

$\xi f_{2}(u)\leq\xi k_{2}(u+1)^{\beta}\leq\frac{\chi
k_{3}}{2}(u+1)^{\alpha}+c\textsubscript{4}\leq\frac{\chi}{2}f_{1}(u)+c\textsubscript{4},$

with $c\textsubscript{4}=\left(\frac{2\beta\xi k_{2}}{\chi
k_{3}\alpha}\right)^{\frac{\alpha}{\alpha-\beta}}\frac{\chi
k_{3}(\alpha-\beta)}{2\beta}>0$, which implies

(54) $\frac{\chi}{2}f_{1}(u)-c\textsubscript{4}\leq f(u)\leq\chi f_{1}(u).$

From the concavity of $U$ in Lemma 6.1, it is seen that $U_{s}$ is
nonincreasing, namely $U_{s}(\sigma,t)\geq U_{s}(s,t)$ for any
$\sigma\in(0,s)$. Henceforth, since $f_{1}$ is nondecreasing (recall (12)), we
have

(55)
$\int_{0}^{s}f_{1}(nU_{s}(\sigma,t))\,d\sigma\geq\int_{0}^{s}f_{1}(nU_{s}(s,t))\,d\sigma=sf_{1}(nU_{s}(s,t))\quad\text{for
all }s\in(0,s_{0})\text{ and }t\in(0,T_{max}).$

Therefore, from (54) and (55) we derive the estimate

$\begin{split}I_{3}&=\frac{1}{2}\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\left[\int_{0}^{s}f(nU_{s}(\sigma,t))\,d\sigma\right]\,ds\\\
&\geq\frac{1}{2}\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\left(\frac{\chi}{2}sf_{1}(nU_{s})-c\textsubscript{4}s\right)\,ds\\\
&\geq\frac{\chi
k_{3}}{4}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1+\alpha}\,ds-\frac{c\textsubscript{4}}{2}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\,ds\quad\text{for
all }t\in S_{\phi}.\end{split}$

Then, for $c_{0}$ as in Lemma 6.1,

(56) $I_{2}+I_{3}\geq\frac{\chi
k_{3}}{4}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1+\alpha}\,ds-\left(c_{0}+\frac{c\textsubscript{4}}{2}\right)\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\,ds\quad\text{for
all }t\in S_{\phi}.$

Now, we focus on the first integral in (56). Since $m_{2}+\alpha-1>0$, clearly
we get that $(nU_{s}+1)^{m_{2}+\alpha-1}>(nU_{s})^{m_{2}+\alpha-1}$, and by
exploiting this latter we have

(57) $\frac{\chi
k_{3}}{4}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1+\alpha}\,ds\geq\frac{\chi
k_{3}n^{m_{2}+\alpha-1}}{4}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}^{m_{2}+\alpha}\,ds=\frac{\chi
k_{3}n^{m_{2}+\alpha-1}}{4}\psi(t)\quad\text{on }S_{\phi}.$

In order to treat the second integral in (56), we use inequality (29): since
$m_{2}>0$, we obtain for
$c\textsubscript{5}=\left(c_{0}+\frac{c\textsubscript{4}}{2}\right)$ the
following estimate

(58)
$\begin{split}&-c\textsubscript{5}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\,ds=-\frac{1}{n}c\textsubscript{5}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)\frac{nU_{s}}{nU_{s}+1}(nU_{s}+1)^{m_{2}}\,ds\\\
&\geq-\frac{1}{n}c\textsubscript{5}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)(nU_{s}+1)^{m_{2}}\,ds\\\
&\geq-c\textsubscript{6}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}^{m_{2}}\,ds-c\textsubscript{7}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)\,ds\quad\text{for
all }t\in S_{\phi}.\end{split}$

Additionally, from $\frac{m_{2}}{m_{2}+\alpha}<1$ we can apply the Hölder
inequality to the first integral in (58), which leads to

(59)
$\begin{split}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}^{m_{2}}\,ds&=\int_{0}^{s_{0}}\left(s^{1-\gamma}(s_{0}-s)U_{s}^{m_{2}+\alpha}\right)^{\frac{m_{2}}{m_{2}+\alpha}}s^{\frac{(1-\gamma)\alpha}{m_{2}+\alpha}}(s_{0}-s)^{\frac{\alpha}{m_{2}+\alpha}}\,ds\\\
&\leq\left(\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}^{m_{2}+\alpha}\,ds\right)^{\frac{m_{2}}{m_{2}+\alpha}}\left(\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)\,ds\right)^{\frac{\alpha}{m_{2}+\alpha}}\\\
&=\psi^{\frac{m_{2}}{m_{2}+\alpha}}(t)\,((2-\gamma)(3-\gamma))^{-\frac{\alpha}{m_{2}+\alpha}}s_{0}^{\frac{(3-\gamma)\alpha}{m_{2}+\alpha}}\quad\text{for
all }t\in S_{\phi}.\end{split}$

By putting (59) into (58), we obtain for every $t\in S_{\phi}$

(60)
$\begin{split}-c\textsubscript{5}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\,ds&\geq-c\textsubscript{6}\psi^{\frac{m_{2}}{m_{2}+\alpha}}(t)\,((2-\gamma)(3-\gamma))^{-\frac{\alpha}{m_{2}+\alpha}}s_{0}^{\frac{(3-\gamma)\alpha}{m_{2}+\alpha}}-\frac{c\textsubscript{7}}{(2-\gamma)(3-\gamma)}s_{0}^{3-\gamma},\end{split}$

so by invoking (56) and taking into account (57) and (60), we can conclude. ∎

In order to obtain the desired superlinear ODI for $\phi$, we have to rely on
some relations involving $U$ and $\psi$ and $\phi.$

######  Lemma 6.5.

Let $u_{0}$ satisfy its related assumptions in (10) and (12), $m_{2},\alpha>0$
and $\gamma\in(-\infty,1)$ be such that

$m_{2}+\alpha>1\quad\text{and}\quad 2-(m_{2}+\alpha)<\gamma<1.$

Then there exist $c\textsubscript{1}$, $c\textsubscript{2}$ such that for any
$s_{0}\in\left(0,\frac{R^{n}}{6}\right]$

(61) $U(s,t)\leq
c\textsubscript{1}s^{\frac{m_{2}+\alpha+\gamma-2}{m_{2}+\alpha}}(s_{0}-s)^{-\frac{1}{m_{2}+\alpha}}\psi^{\frac{1}{m_{2}+\alpha}}(t)\quad\text{for
all }s\in(0,s_{0})\text{ and }t\in S_{\phi},$

and

(62) $\psi(t)\geq
c\textsubscript{2}s_{0}^{-(3-\gamma)(m_{2}+\alpha-1)}\phi^{m_{2}+\alpha}(t)\quad\text{for
all }t\in S_{\phi}.$

###### Proof.

The proof of (61) and (62) follows by [Tan22, Lemma 3.8] and [WL21, Lemma
3.7], respectively. ∎

The following is precisely the lemma relying on assumption
($\mathcal{H}_{4}$).

######  Lemma 6.6.

Under the same assumptions of Lemma 6.1, let $m_{1}\in\mathbb{R}$,
$m_{2},\alpha,\beta>0$ and $k>1$ be such that constrains ($\mathcal{H}_{4}$)
in Assumptions 2.1 are satisfied. Then there exist $\varepsilon>0$ small
enough, $\gamma=\gamma(m_{1},m_{2},\alpha,k)\in(-\infty,1)$ and
$c\textsubscript{1}$, $c\textsubscript{2}$, $c\textsubscript{3}$,
$c\textsubscript{4}$, $c\textsubscript{5}$ and $c\textsubscript{6}$ such that
for $s_{0}\in\left(0,\frac{R^{n}}{6}\right]$ one has

$\phi^{\prime}(t)\geq\begin{cases}c\textsubscript{1}s_{0}^{-(3-\gamma)(m_{2}+\alpha-1)}\phi^{m_{2}+\alpha}(t)-c\textsubscript{2}s_{0}^{3-\gamma-\frac{2}{n}\frac{(m_{2}+\alpha)}{m_{2}+\alpha-
m_{1}}}&\text{if }m_{1}>0,\\\
c\textsubscript{3}s_{0}^{-(3-\gamma)(m_{2}+\alpha-1)}\phi^{m_{2}+\alpha}(t)-c\textsubscript{4}s_{0}^{3-\gamma-\frac{2}{n}\frac{(m_{2}+\alpha)}{m_{2}+\alpha-\epsilon}}&\text{if
}m_{1}=0,\\\
c\textsubscript{5}s_{0}^{-(3-\gamma)(m_{2}+\alpha-1)}\phi^{m_{2}+\alpha}(t)-c\textsubscript{6}s_{0}^{3-\gamma-\frac{2}{n}}&\text{if
}m_{1}<0,\end{cases}\quad\quad\textrm{for all }t\in S_{\phi}.$

###### Proof.

By substituting the estimates of $I_{1},I_{2},I_{3},I_{4}$ given in Lemmas
6.3, 6.4 and 6.5 into Lemma 6.2, we can adapt [Tan22, Lemma 3.10] taking into
account that [Tan22, (3.12), Lemma 3.5] is replaced by (53), so being
necessary manipulating the term involving
$\psi^{\frac{m_{2}}{m_{2}+\alpha}}(t)s_{0}^{(3-\gamma)\frac{\alpha}{m_{2}+\alpha}}$
by the Young inequality and relation (62). ∎

The previous lemmata allows us to conclude.

#### Proof of Theorem 2.3:

We focus only on the situation where $m_{1}>0$, the cases $m_{1}=0$ and
$m_{1}<0$ being similar. Since ($\mathcal{H}_{4}$) holds, we can apply Lemma
6.6 and find $\gamma\in(-\infty,1)$, $c\textsubscript{7},c\textsubscript{8}$
such that for each $u_{0}$ fulfilling the related restrictions in (10) and
(12), and $s_{0}\leq\frac{R^{n}}{6}$, one deduces

(63) $\phi^{\prime}(t)\geq
c\textsubscript{7}s_{0}^{-(3-\gamma)(m_{2}+\alpha-1)}\phi^{m_{2}+\alpha}(t)-c\textsubscript{8}s_{0}^{3-\gamma-\frac{2}{n}\frac{(m_{2}+\alpha)}{m_{2}+\alpha-
m_{1}}}\quad\text{for all }t\in S_{\phi}.$

Next we pick $s_{0}\leq\frac{R^{n}}{6}$ small enough such that

(64) $s_{0}\leq\frac{M}{2}$

and

(65) $s_{0}^{m_{2}+\alpha-\frac{2}{n}\frac{(m_{2}+\alpha)}{m_{2}+\alpha-
m_{1}}}\leq\frac{c\textsubscript{7}}{2c\textsubscript{8}}\left(\frac{M}{2(1-\gamma)(2-\gamma)\omega_{n}}\right)^{m_{2}+\alpha}.$

Moreover, from

$\frac{M_{0}-\epsilon_{0}}{\omega_{n}}\int_{s_{*}}^{s_{0}}s^{-\gamma}(s_{0}-s)\,ds=\frac{M_{0}-\epsilon_{0}}{\omega_{n}}\left[\frac{s_{0}^{2-\gamma}}{(1-\gamma)(2-\gamma)}-s_{*}^{1-\gamma}\frac{s_{0}}{1-\gamma}+\frac{s_{*}^{2-\gamma}}{2-\gamma}\right],$

we can take $\epsilon_{0}\in\left(0,\frac{s_{0}}{2}\right)$ and
$s_{*}\in\left(0,s_{0}\right)$ so small and satisfying

(66)
$\frac{M_{0}-\epsilon_{0}}{\omega_{n}}\int_{s_{*}}^{s_{0}}s^{-\gamma}(s_{0}-s)\,ds>\frac{M_{0}-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma}.$

We set $r_{*}:=s_{*}^{\frac{1}{n}}\in(0,R)$ and require to $u_{0}$ to satisfy
(16). In order to show that $T_{max}<\infty$, we argue by establishing that if
by contradiction $T_{max}=\infty$, then
$\tilde{T}=\sup\tilde{S}\in(0,\infty]$, with

(67) $\tilde{S}:=\left\\{T>0\textrm{ such that
}\phi(t)>\frac{M-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma}\quad\text{for
all }t\in[0,T]\right\\},$

would be at the same time finite and infinite.

First, by observing that
$\phi(0)>\frac{M-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma}$, the
continuity of $\phi$ ensures that $\tilde{S}$ is not empty. Indeed, we have
that for any $s\in(s_{*},R^{n})$

$U(s,0)\geq
U(s_{*},0)=\frac{1}{\omega_{n}}\int_{B_{r_{*}}(0)}u_{0}\,dx\geq\frac{M_{0}-\epsilon_{0}}{\omega_{n}}.$

In turn, due to $M=\max\\{M_{0},C\\}=M_{0}$ thanks to the assumption
$M_{0}>C$, we deduce from (66) that

$\begin{split}\phi(0)&\geq\int_{s_{*}}^{s_{0}}s^{-\gamma}(s_{0}-s)U(s,0)\,ds\geq\frac{M_{0}-\epsilon_{0}}{\omega_{n}}\int_{s_{*}}^{s_{0}}s^{-\gamma}(s_{0}-s)\,ds\\\
&>\frac{M_{0}-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma}=\frac{M-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma}.\end{split}$

Now, by exploiting (67) and (64) leads to

$\phi(t)\geq\frac{M}{2(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma}\quad\text{for
all }t\in(0,\tilde{T}),$

so that $(0,\tilde{T})\subset S_{\phi}$ (recall (50)).

Secondly, condition (65) implies

$\frac{\frac{c\textsubscript{7}}{2}s_{0}^{-(3-\gamma)(m_{2}+\alpha-1)}\phi^{m_{2}+\alpha}(t)}{c\textsubscript{8}s_{0}^{3-\gamma-\frac{2}{n}\frac{(m_{2}+\alpha)}{m_{2}+\alpha-
m_{1}}}}\geq\frac{c\textsubscript{7}}{2c\textsubscript{8}}\left(\frac{M}{2(1-\gamma)(2-\gamma)\omega_{n}}\right)^{m_{2}+\alpha}s_{0}^{-(m_{2}+\alpha)+\frac{2}{n}\frac{(m_{2}+\alpha)}{m_{2}+\alpha-
m_{1}}}\geq 1\quad\textrm{on}\;(0,\tilde{T}),$

which subsequently gives from (63)

(68)
$\phi^{\prime}(t)\geq\frac{c\textsubscript{7}}{2}s_{0}^{-(3-\gamma)(m_{2}+\alpha-1)}\phi^{m_{2}+\alpha}(t)\geq
0\quad\text{for all }t\in(0,\tilde{T}).$

With these facts, let us now establish the inconsistency $\tilde{T}<\infty$
and $\tilde{T}=\infty$. By integrating inequality (68) on $(0,\tilde{T})$, we
have that

$\int_{0}^{\tilde{T}}\left(\frac{1}{1-(m_{2}+\alpha)}\phi^{1-(m_{2}+\alpha)}(t)\right)^{\prime}\,dt\geq\int_{0}^{\tilde{T}}\frac{c\textsubscript{7}}{2}s_{0}^{-(3-\gamma)(m_{2}+\alpha-1)}\,dt,$

so that due to $m_{2}+\alpha-1>0$ and the nonnegative property of $\phi$

$\frac{c\textsubscript{7}}{2}s_{0}^{-(3-\gamma)(m_{2}+\alpha-1)}\tilde{T}\leq\frac{\phi^{1-(m_{2}+\alpha)}(\tilde{T})}{1-(m_{2}+\alpha)}-\frac{\phi^{1-(m_{2}+\alpha)}(0)}{1-(m_{2}+\alpha)}\leq\frac{\phi^{1-(m_{2}+\alpha)}(0)}{m_{2}+\alpha-1},$

or explicitly

$\tilde{T}\leq\frac{2}{c\textsubscript{7}(m_{2}+\alpha-1)\phi^{m_{2}+\alpha-1}(0)}s_{0}^{(3-\gamma)(m_{2}+\alpha-1)}<\infty.$

Nevertheless, we have to exclude the finiteness of $\tilde{T}=\sup\tilde{S}$;
in fact, by the definition of $\tilde{S}$ in (67), we should have

(69)
$\phi(\tilde{T})=\frac{M-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma},$

because if

$\phi(\tilde{T})>\frac{M-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma},$

by continuity of $\phi$ we would have that $\tilde{T}$ cannot be the supremum
of $\tilde{S}$. But (69) cannot be true since from the nondecreasing of $\phi$
in view of inequality (68), we would arrive at

$\frac{M-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma}=\phi(\tilde{T})\geq\phi(0)>\frac{M-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}s_{0}^{2-\gamma}.$

As a conclusion, the constructed $u_{0}$ implies that $T_{max}$ has to be
finite. ∎

######  Remark 4 (Finite-time blow-up with $m_{2}\neq m_{3}$).

As already said in $\S$3, the proof of the blow-up for problem (8) with
$m_{2}\neq m_{3}$ is still an open problem. Since the transformation $z=\chi
v-\xi w$ used above to reorganize problem (8) with $m_{2}=m_{3}$ into the
simplified version (42) is not longer employable, by reasoning as in Lemma
6.2, the corresponding inequality (52) would read

$\begin{split}\phi^{\prime}(t)&\geq
n^{2}\int_{0}^{s_{0}}s^{2-\frac{2}{n}-\gamma}(s_{0}-s)(nU_{s}+1)^{m_{1}-1}U_{ss}\,ds-\chi
f_{1_{\gamma}}\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\,ds\\\
&\quad+\frac{\chi}{2}\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{2}-1}\left[\int_{0}^{s}f_{1}(nU_{s}(\sigma,t))\,d\sigma\right]\,ds-\mu
n^{k-1}\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)\left[\int_{0}^{s}U_{s}^{k}(\sigma,t)\,d\sigma\right]\,ds\\\
&\quad+\xi\int_{0}^{s_{0}}s^{1-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{3}-1}m_{2}(t)\,ds-\xi\int_{0}^{s_{0}}s^{-\gamma}(s_{0}-s)U_{s}(nU_{s}+1)^{m_{3}-1}\left[\int_{0}^{s}f_{2}(nU_{s}(\sigma,t))\,d\sigma\right]\,ds,\end{split}$

valid for all $t\in S_{\phi}$ and with
$f_{1_{\gamma}}=f_{1}\left(\frac{8n}{2^{\gamma}(3-\gamma)\omega_{n}}\right)$.
In particular, the extra terms involving the repulsion coefficient $\xi$ make
the analysis more complex. This is also connected to the signs of such terms
(exactly opposite than those of the integrals associated to the attraction
coefficient $\chi$), which do not allow to use the right inequalities tied to
the hypotheses on $f_{i}$, and in turn on $m_{i}$.

#### Acknowledgements

The authors 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), and are partially supported by the research projects
_Evolutive and Stationary Partial Differential Equations with a Focus on
Biomathematics_ (2019, Grant Number: F72F20000200007), Analysis of PDEs in
connection with real phenomena (2021, Grant Number: F73C22001130007), funded
by Fondazione di Sardegna. GV is also supported by MIUR (Italian Ministry of
Education, University and Research) Prin 2017 _Nonlinear Differential Problems
via Variational, Topological and Set-valued Methods_ (Grant Number:
2017AYM8XW).

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