Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T02:35:05.085Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bilotta, E., Gargano, F., Giunta, V., Lombardo, M. C., Pantano, P. and Sammartino, M.. Eckhaus and zigzag instability in a chemotaxis model of multiple sclerosis. Atti della Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche, Matematiche e Naturali 96 (2018), 9.Google Scholar
Bilotta, E., Gargano, F., Giunta, V., Lombardo, M. C., Pantano, P. and Sammartino, M.. Axisymmetric solutions for a chemotaxis model of Multiple Sclerosis. Ricerche di Matematica 68 (2019), 281–294.CrossRefGoogle Scholar
Calvez, V. and Khonsari, R. H.. Mathematical description of concentric demyelination in the human brain: self-organization models, from liesegang rings to chemotaxis. Math. Comput. Model 47 (2008), 726–742.CrossRefGoogle Scholar
Cañizo, J. A., Desvillettes, L. and Fellner, K.. Improved duality estimates and applications to reaction-diffusion equations. Commun. Partial Differential Equations 39 (2014), 1185–1204.CrossRefGoogle Scholar
Desvillettes, L. and Giunta, V.. Existence and regularity for a chemotaxis model involved in the modeling of multiple sclerosis. Ricerche di Matematica 70 (2021), 99–113.CrossRefGoogle Scholar
Desvillettes, L., Lepoutre, T., Moussa, A. and Trescases, A.. On the entropic structure of reaction-cross diffusion systems. Commun. Partial Differential Equations 40 (2015), 1705–1747.CrossRefGoogle Scholar
Desvillettes, L. and Trescases, A.. New results for triangular reaction cross diffusion systems. J. Math. Anal. Appl. 430 (2015), 32–59.CrossRefGoogle Scholar
Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order (Berlin: Springer, 2015).Google Scholar
Hu, X., Fu, S. and Ai, S.. Global asymptotic behavior of solutions for a parabolic-parabolic-ode chemotaxis system modeling multiple sclerosis. J. Differ. Equ. 269 (2020), 6875–6898.CrossRefGoogle Scholar
Khonsari, R. H. and Calvez, V.. The origins of concentric demyelination: self-organization in the human brain. PLoS ONE 2 (2007), e150.CrossRefGoogle ScholarPubMed
Ladyženskaija, O. A., Alekseevich Solonnikov, V. and Uralceva, N. N.. Linear and quasilinear equations of parabolic type, volume 23. American Mathematical Soc., 1988.Google Scholar
Lombardo, M. C., Barresi, R., Bilotta, E., Gargano, F., Pantano, P. and Sammartino, M.. Demyelination patterns in a mathematical model of multiple sclerosis. J. Math. Biol. 75 (2017), 373–417.CrossRefGoogle Scholar
Morgan, J. and Quoc Tang, B.. Boundedness for reaction–diffusion systems with lyapunov functions and intermediate sum conditions. Nonlinearity 33 (2020), 3105.CrossRefGoogle Scholar
Pierre, M.. Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78.2 (2010), 417–455.CrossRefGoogle Scholar
Simon, J.. Compact sets in the space $L^{p}(0,T;B)$. Annali di Matematica pura ed applicata 146 (1986), 65–96.CrossRefGoogle Scholar
Winkler, M.. Aggregation vs. global diffusive behavior in the higher-dimensional keller–segel model. J. Differ. Equ. 248 (2010), 2889–2905.CrossRefGoogle Scholar