Dissipation Through Combinations of Nonlocal and Gradient Nonlinearities in Chemotaxis Models
Authors: Díaz Fuentes Rafael , Frassu Silvia , Viglialoro Giuseppe 
Published: February 2025

This work concerns with a class of chemotaxis models in which external sources, comprising nonlocal and gradient-dependent damping reactions, influence the motion of a cell density attracted by a chemical signal. The mechanism of the two densities is studied in bounded and impenetrable regions. In particular, it is seen that no gathering effect for the cells can appear in time provided that the damping impacts are sufficiently strong. Mathematically, we study this problem

$$ \textstyle\begin{cases} u_{t}=\nabla \cdot \left ((u+1)^{m_{1}-1}\nabla u -\chi u(u+1)^{m_{2}-1} \nabla v\right )+ B(u,\nabla u)&{\mathrm{in}}\ \Omega \times \{t>0\} , \\ \tau v_{t}=\Delta v-v+f(u) &{\mathrm{in}}\ \Omega \times \{t>0\}, \\ u_{\nu }=v_{\nu }=0 &{\mathrm{on}}\ \partial \Omega \times \{t>0\}, \\ u(x, 0)=u_{0}(x), \tau v(x,0)= \tau v_{0}(x) &x \in \bar{\Omega }, \end{cases} $$

{

u
t

=
∇
⋅

(

(
u
+
1
)

m
1

−
1

∇
u
−
χ
u

(
u
+
1
)

m
2

−
1

∇
v
)

+
B
(
u
,
∇
u
)

in

Ω
×
{
t
>
0
}
,

τ

v
t

=
Δ
v
−
v
+
f
(
u
)

in

Ω
×
{
t
>
0
}
,

u
ν

=

v
ν

=
0

on

∂
Ω
×
{
t
>
0
}
,

u
(
x
,
0
)
=

u
0

(
x
)
,
τ
v
(
x
,
0
)
=
τ

v
0

(
x
)

x
∈

Ω
¯

,

for

$$ B(u,\nabla u)=B \textrm{ being either \; } au^{\alpha }-b u^{\beta }-c \int _{\Omega }u^{\delta }, \textrm{ or \; } au^{\alpha }-b u^{\alpha }\int _{\Omega }u^{\beta }-c|\nabla u|^{\delta }, $$

B
(
u
,
∇
u
)
=
B
 being either  
a

u
α

−
b

u
β

−
c

∫
Ω

u
δ

,
 or  
a

u
α

−
b

u
α

∫
Ω

u
β

−
c
|
∇
u

|
δ

,

and where

$\Omega $

Ω

is a bounded and smooth domain of

$\mathbb{R}^{n}$

R
n

(

$n \in \mathbb{N}$

n
∈
N

),

$\{t>0\}\subseteq (0,\infty )$

{
t
>
0
}
⊆
(
0
,
∞
)

an open interval,

$\tau \in \{0,1\}$

τ
∈
{
0
,
1
}

,

$m_{1},m_{2}\in \mathbb{R}$

m
1

,

m
2

∈
R

,

$\chi ,a,b>0$

χ
,
a
,
b
>
0

,

$c\geq 0$

c
≥
0

, and

$\alpha , \beta ,\delta \geq 1$

α
,
β
,
δ
≥
1

. Herein for

$(x,t)\in \Omega \times \{t>0\}$

(
x
,
t
)
∈
Ω
×
{
t
>
0
}

,

$u=u(x,t)$

u
=
u
(
x
,
t
)

stands for the population density,

$v=v(x,t)$

v
=
v
(
x
,
t
)

for the chemical signal and

$f$

f

for a regular function describing the production law. The population density and the chemical signal are initially distributed accordingly to nonnegative and sufficiently regular functions

$u_{0}(x)$

u
0

(
x
)

and

$\tau v_{0}(x)$

τ

v
0

(
x
)

, respectively. For each of the expressions of

$B$

B

, sufficient conditions on parameters of the models ensuring that any nonnegative classical solution

$(u,v)$

(
u
,
v
)

to system (◊) is such that

$\{t>0\} \equiv (0,\infty )$

{
t
>
0
}
≡
(
0
,
∞
)

and uniformly bounded in time, are established. In the literature, most of the results concerning chemotaxis models with external sources deal with classical logistics, for which

$B=a u^{\alpha }-b u^{\beta }$

B
=
a

u
α

−
b

u
β

. Thereafter, the introduction of dissipative effects as those expressed in

$B$

B

is the main novelty of this investigation. On the other hand, this paper extends the analyses in (Chiyo et al. in Appl. Math. Optim. 89(9):1–21, 2024; Bian et al. in Nonlinear Anal. 176:178–191, 2018; Latos in Nonlocal reaction preventing blow-up in the supercritical case of chemotaxis, 2020, arXiv:2011.10764).