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On a repulsion Keller–Segel system with a logarithmic sensitivity

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

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: [email protected]; [email protected]
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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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