Applied Mathematics, 2020, 11, 985-990
https://www.scirp.org/journal/am
ISSN Online: 2152-7393
ISSN Print: 2152-7385
DOI: 10.4236/am.2020.1110064 Oct. 19, 2020 985 Applied Mathematics
Global Bounded Solutions for the
Keller-Segel Chemotaxis System
with Singular Sensitivity
Khalid Ahmed Abbakar
1,2*
, Omer Khalil
1
, Alhussein Mohamed
1
, Bechir Mahamat
1
,
Abdoulaye Ali
1
, Abeer Alhadi
3
1
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
2
Department of Mathematics and Physics, Faculty of Education, University of Gadarif, Gadarif, Sudan
3
College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou, China
Abstract
In this paper, we consider the Neumann initial-boundary value problem for
the Keller-Segel chemotaxis system with singular sensitivity
1
2
,
,
t
t
u
u d u v
v
v d v v u
χ
= ∆ − ∇⋅ ∇
= ∆− +
(0.1)
is considered in a bounded domain with smooth boundary, ( ) 1
n
n Ω⊂ ≥ ,
where
1 2
0, 0 d d > > with parameter χ ∈ . When
1 2
d d χ = + , satisfying
for all initial data
( )
0
0
0 u C ≤ ∈ Ω and ( )
1,
0
0 v W
∞
< ∈ Ω , we prove that the
problem possesses a unique global classical solution which is uniformly
bounded in ( ) 0, Ω× ∞ .
Keywords
Keller-Segel System, Chemotaxis, Global Bounded Solution, Singular
Sensitivity
1. Introduction
The Keller-Segel system is used to model chemotactic movement in biology [1].
The mathematical study of the system has attracted great interest in recent years
[2]. In this paper, we consider the Neumann initial-boundary value problem for
the chemotaxis system with singular sensitivity
How to cite this paper: Abbakar, K.A.,
Khalil, O., Mohamed, A., Mahamat, B., Ali,
A. and Alhadi, A. (2020) Global Bounded
Solutions for the Keller-Segel Chemotaxis
System with Singular Sensitivity. Applied
Mathematics, 11, 985-990.
https://doi.org/10.4236/am.2020.1110064
Received: September 17, 2020
Accepted: October 16, 2020
Published: October 19, 2020
Copyright © 2020 by author(s) and
Scientific Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access