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Global well-posedness and nonlinear stability of a chemotaxis system modelling multiple sclerosis

Part of: Partial differential equations Physiological, cellular and medical topics Parabolic equations and systems

Published online by Cambridge University Press:  28 July 2021

Laurent Desvillettes ,
Valeria Giunta ,
Jeff Morgan  and
Bao Quoc Tang
Laurent Desvillettes
Affiliation:
Université de Paris, Sorbonne Université, CNRS Institut de Mathématiques de Jussieu-Paris Rive Gauche, Paris, France (desvillettes@math.univ-paris-diderot.fr)
Valeria Giunta
Affiliation:
Department of Engineering, University of Palermo, Palermo, Italy (valeria.giunta@unipa.it)
Jeff Morgan
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77004, USA (jmorgan@math.uh.edu)
Bao Quoc Tang
Affiliation:
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria (quoc.tang@uni-graz.at, baotangquoc@gmail.com)
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Abstract

We consider a system of reaction–diffusion equations including chemotaxis terms and coming out of the modelling of multiple sclerosis. The global existence of strong solutions to this system in any dimension is proved, and it is also shown that the solution is bounded uniformly in time. Finally, a nonlinear stability result is obtained when the chemotaxis term is not too big. We also perform numerical simulations to show the appearance of Turing patterns when the chemotaxis term is large.

Keywords

Chemotaxis modelsGlobal solutionsUniform-in-time boundsNonlinear stabilityCross diffusion

MSC classification

Primary: 35K51: Initial-boundary value problems for second-order parabolic systems
Secondary: 35K57: Reaction-diffusion equations 35K92: Quasilinear parabolic equations with $p$-Laplacian 92C17: Cell movement (chemotaxis, etc.) 35B40: Asymptotic behavior of solutions

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 826 - 856
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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