On a Parabolic-ODE chemotaxis system with periodic asymptotic behavior
Authors: Negreanu M. , Tello J. , Vargas A. 
Published: October 2023

We consider a system of differential equations modeling chemotaxis, the habillity of some living organisms to move towards a higher concentration of a chemical signal. The system consists of two differential equations, a parabolic one describing the behavior of a biological species “

u
u

” coupled to second equation modeling the concentration of a chemical substance “

v
v

”. The growth of the biological species is limited by a logistic-like term where the carrying capacity presents a time-periodic asymptotic behavior. The production of the chemical species is described in terms of a regular function

h
h

, which increases as “

u
u

” increases. The system is presented in a regular bounded domain

Ω
⊂

R

n

\Omega \subset \mathbb {R}^n

, with positive constant chemotaxis coefficient

χ
\chi

in the following way
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row u Subscript t Baseline equals normal upper Delta u minus d i v left-parenthesis chi u nabla v right-parenthesis plus mu u left-parenthesis 1 minus u plus f right-parenthesis comma x element-of normal upper Omega comma t greater-than 0 comma 2nd Row epsilon v Subscript t Baseline minus upper D Subscript v Baseline normal upper Delta v equals h left-parenthesis u comma v right-parenthesis comma x element-of normal upper Omega comma t greater-than 0 comma EndLayout">

{

u
t

=
Δ

u

−
d
i
v
(
χ
u
∇

v

)
+
μ
u
(
1
−
u
+
f
)
,

x
∈
Ω
,

t
>
0
,

ϵ

v
t

−

D
v

Δ
v
=
h
(
u
,
v
)
,

x
∈
Ω
,

t
>
0
,

\begin{equation*} \begin {cases} u_t = \Delta {u} - div(\chi u \nabla {v}) + \mu u(1-u+f), \quad x\in \Omega , \; t>0, \\[2mm] \epsilon v_t- D_v \Delta v= h(u,v) , \quad x\in \Omega , \; t>0, \end{cases} \end{equation*}

with initial data

(

u
0

,

v
0

)

(u_0, v_0)

and appropiate boundary conditions for

u
u

. The function

f
f

, in the reaction term, is a bounded given function fulfilling
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f left-parenthesis x comma t right-parenthesis minus f Superscript asterisk Baseline left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis normal upper Omega right-parenthesis Baseline right-arrow 0 comma as t right-arrow normal infinity comma">

‖
f
(
x
,
t
)
−

f
∗

(
t
)

‖

L

∞

(
Ω
)

→
0
,

 as 

t
→
∞
,

\begin{equation*} \|f(x,t)-f^*(t)\|_{L^{\infty }(\Omega )}\rightarrow 0, \quad \text { as }\quad t\rightarrow \infty , \end{equation*}

with

f
∗

(
t
)

f^*(t)

being a time-periodic function independent of the space variable “

x
x

”.

Three different cases may occur:

If in the equation of

v
v

, the diffusion is dominant in the time scale we are working, then, the system is simplify to a Parabolic-Elliptic equation

If there is not diffusion of

v
v

, the problem is a Parabolic-ODE system.

When diffusion is not dominant and neither neglectable in the time scales we are studying the the problem.

In the chapter we present results of existence of solutions and its asymptotic behavior under suitable assumptions on the initial data for given functions

f
f

and

h
h

.