Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T10:31:52.377Z Has data issue: false hasContentIssue false
  • English
  • Français

On a repulsion Keller–Segel system with a logarithmic sensitivity

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

Published online by Cambridge University Press:  14 January 2021

JIE JIANG Open the ORCID record for JIE JIANG [Opens in a new window]
JIE JIANG*
Affiliation:
Innovation Academy for Precision Measurement Science and Technology, CAS, Wuhan, Hubei Province 430071, P.R. China emails: jiang@wipm.ac.cn; jiang@apm.ac.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the initial-boundary value problem of a repulsion Keller–Segel system with a logarithmic sensitivity modelling the reinforced random walk. By establishing an energy–dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoy an eventual regularity property, i.e., it becomes regular after certain time T > 0. An exponential convergence rate towards the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.

Keywords

Chemotaxisglobal existencerepulsionlogarithmic sensitivityeventual regularity

MSC classification

Primary: 35B35: Stability 35D30: Weak solutions
Secondary: 35K59: Quasilinear parabolic equations 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations 92C17: Cell movement (chemotaxis, etc.)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 1 , February 2022 , pp. 153 - 181
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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

Ahn, J. & Yoon, C. (2019) Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing. Nonlinearity 32, 1327–1351.CrossRefGoogle Scholar
Alikakos, N. (1979) An application of the invariance principle to reaction-diffusion equations. J. Diff. Equ. 33, 201–225.CrossRefGoogle Scholar
Bellomo, N., Belouquid, A., Tao, Y. & Winkler, M. (2015) Toward a mathematical theory of Keller–Segel models of pattern formation in biology tissues. Math. Mod. Meth. Appl. Sci. 25, 1663–1763.CrossRefGoogle Scholar
Black, T. (2018) Eventual smoothness of generalized solutions to a singular chemotaxis–Stokes system in 2D. J. Differ. Equ. 265, 2296–2339.CrossRefGoogle Scholar
Cao, X. (2015) Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete Contin. Dyn. Syst. Ser. A 35, 1891–1904.CrossRefGoogle Scholar
Cieślak, T., Laurençot, P. & Morales-Rodrigo, C. (2008) Global existence and convergence to steady states in a chemorepulsion system. Banach Center Publ. Polish Acad. Sci., Warsaw 81, 105–117.CrossRefGoogle Scholar
Fujie, K. & Senba, T. (2016) Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity. Nonlinearity 29, 2417–2450.CrossRefGoogle Scholar
Fujie, K. & Senba, T. (2018) A sufficient condition of sensitivity functions for boundedness of solutions to a parabolic-parabolic chemotaxis system. Nonlinearity 31, 1639–1672.CrossRefGoogle Scholar
Fujie, K., Winkler, M. & Yokota, T. (2014) Blow-up prevention by a logistic sources in a parabolic-elliptic Keller–Segel system with singular sensitivity. Nonlinear Anal. 109, 56–71.CrossRefGoogle Scholar
Hillen, T. & Painter, K. (2009) A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217.CrossRefGoogle ScholarPubMed
Horstmann, D. & Wang, G.-F. (2001) Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12, 159–177.CrossRefGoogle Scholar
Horstmann, D. & Winkler, M. (2005) Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107.CrossRefGoogle Scholar
Jiang, J., Wu, H. & Zheng, S. (2015) Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains. Asymptotic Anal. 92, 249–258.CrossRefGoogle Scholar
Jiang, J., Wu, H. & Zheng, S. (2018) Blow-up for a three dimensional Keller–Segel model with consumption of chemoattractant. J. Differ. Equ. 264, 5432–5464.CrossRefGoogle Scholar
Jiang, J. (2020) Global stability of Keller–Segel systems in critical Lebesgue spaces. Discrete Cont. Dyn. Syst. A 40, 609–634.CrossRefGoogle Scholar
Jiang, J. (2019) Global stability of homogeneous steady states in scaling-invariant spaces for a Keller–Segel–Navier–Stokes system. J. Differ. Equ. 267, 659–692.CrossRefGoogle Scholar
Jiang, J. (2019) Eventual smoothness and exponential stabilization of global weak solutions to some chemotaxis systems. SIAM J. Math. Anal. 51, 4604–4644.CrossRefGoogle Scholar
Kozono, H., Miura, M. & Sugiyama, Y. (2016) Existence and uniqueness theorem on mild solutions to the Keller–Segel system coupled with the Navier–Stokes fluid. J. Funct. Anal. 270, 1663–1683.CrossRefGoogle Scholar
Lankeit, J. (2015) Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source. J. Differ. Equ. 258, 1158–1191.CrossRefGoogle Scholar
Levine, H. A. & Sleeman, B. D. (1997) A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57, 683–730.Google Scholar
Mizoguchi, N. & Souplet, Ph. (2014) Nondegeneracy of blow-up points for the parabolic Keller–Segel system. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 851–875.CrossRefGoogle Scholar
Mock, M. S. (1974) An initial value problem from semiconductor device theory. SIAM J. Math. Anal. 5, 597–612.CrossRefGoogle Scholar
Mock, M. S. (1975) Asymptotic behavior of solutions of transport equations for semiconductor devices. J. Math. Anal. Appl. 49, 215–225.CrossRefGoogle Scholar
Nagai, T. & Senba, T. (1998) Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis. Adv. Math. Sci. Appl. 8, 145–156.Google Scholar
Othmer, H. G. & Stevens, A. (1997) Aggregation, blowup and collapse: The ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1044–1081.Google Scholar
Simon, J. (1987) Compact sets in the space LP(0, T, B). Ann. Mat. Pura Appl. 146, 65–96.CrossRefGoogle Scholar
Tao, Y., Wang, L. & Wang, Z. (2013) Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension. Discrete Contin. Dyn. Syst. Ser. B 18, 821–845.Google Scholar
Tao, Y. (2013) Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete Contin. Dyn. Syst. Ser. B 18, 2705–2722.Google Scholar
Tao, Y. & Winkler, M. (2012) Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equ. 252, 2520–2543.CrossRefGoogle Scholar
Tuval, I., Cisnerous, L., Dombrowski, C., Wolgemuth, C. W., Kessler, J. O. & Goldstein, R. E. (2005) Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. 102, 2277–2282.CrossRefGoogle ScholarPubMed
Winkler, M. (2010) Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905.CrossRefGoogle Scholar
Winkler, M. (2012) Global large-data solutions in a chemotaxis–(Navier–)Stokes system modeling cellular swimming in a fluid drops. Commun. Partial Differ. Equ. 37, 319–351.CrossRefGoogle Scholar
Winkler, M. (2013) Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller–Segel system. J. Math. Pures Appl. 100, 748–767.CrossRefGoogle Scholar
Winkler, M. (2014) Stabilization in a two-dimensional chemotaxis–Navier–Stokes system. Arch. Ration. Mech. Anal. 211, 455–487.Google Scholar
Winkler, M. (2017) How far do chemotaxis-driven forces influence regularity in the Navier–Stokes system? Trans. Am. Math. Soc. 369, 3067–3125.CrossRefGoogle Scholar
Winkler, M. (2019) How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic–elliptic cases. Math. Ann. 373, 1237–1282.CrossRefGoogle Scholar
Winkler, M. (2019) Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis(-Stokes) systems? Int. Math. Res. Not. https://doi.org/10.1093/imrn/rnz056.CrossRefGoogle Scholar
Wrzosek, D. (2004) Global attractor for a chemotaxis model with prevention of overcrowding. Nonlinear Anal. 59, 1293–1310.CrossRefGoogle Scholar
Zheng, S. (2004) Nonlinear Evolution Equations, Chapman & Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar