Keller-Segel Chemotaxis Models: A Review
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Keller-Segel Chemotaxis Models: A Review Gurusamy Arumugam1
· Jagmohan Tyagi1
Received: 12 November 2019 / Accepted: 9 October 2020 © The Author(s), under exclusive licence to Springer Nature B.V. part of Springer Nature 2020
Abstract We recount and discuss some of the most important methods and blow-up criteria for analyzing solutions of Keller-Segel chemotaxis models. First, we discuss the results concerning the global existence, boundedness and blow-up of solutions to parabolic-elliptic type models. Thereafter we describe the global existence, boundedness and blow-up of solutions to parabolic-parabolic models. The numerical analysis of these models is still at a rather early stage only. We recollect quite a few of the known results on numerical methods also and direct the attention to a number of open problems in this domain. Mathematics Subject Classification (2010) 35A01 · 35D30 · 35B44 · 34H15 · 35K40 · 65N30 · 65M08 · 65M06 Keywords Chemotaxis · Keller-Segel models · Weak solutions · Renormalized solutions · Local existence · Global existence · Blow-up · Boundedness · Stabilization · Asymptotic behavior of solutions · Finite difference method · Finite element method · Finite Volume method · Discontinuous Galerkin method
1 Introduction Response to the environmental changes is an essential and basic property of the living cells. Through evolution, both unicellular and multicellular organisms develop various mechanisms that help them to regulate their cellular function in response to environmental changes. In general, whole organisms or cells cannot move by random manner, but they sense their The second author thanks CSIR for the financial support under the grant No. 25(0263)/17/EMR-II and partly DST-SERB under the grant MTR/2018/00096.
B G. Arumugam
[email protected] J. Tyagi [email protected]
1
Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Gandhinagar-382355, Gujarat, India
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environment and respond to it. Movement of the cells is mainly decided by some external stimulants / signals, which determine the direction and distance of the cell movement. In the context of individual cells, migration can be regulated by various environmental factors such as light, chemicals, temperature, electric field, environmental gravity and many more. Of these, chemotaxis is an important sensory phenomenon of the cells by which cells translate chemical signals into motile behavior. Prokaryotic response to chemotaxis have been studied extensively in Escherichia coli, where a quite simple signaling cascade supports the clockwise or anticlockwise rotation of flagella to produce either forward motion or headlong, respectively. In case of eukaryotic organisms, chemotaxis mechanisms have been studied widely in amoeboid Dictyostelium discoideum and mammalian neutrophils in which chemoattractants induce intricate signaling cascades contributing to diverse cellular processes including the establishment of cellular polarity and extension of the cell membrane. Chemo
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