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An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization

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

Published online by Cambridge University Press:  29 May 2020

Wenbin Lv  and
Qingyuan Wang
Wenbin Lv*
Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan030006, China ([email protected])
Qingyuan Wang
Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan030006, China ([email protected])
*
*Corresponding author
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Abstract

This paper deals with the global existence for a class of Keller–Segel model with signal-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain, which can be written as

$$\eqalign{& u_t = \Delta (\gamma (v)u) + \rho u-\mu u^l,\quad x\in \Omega ,\;t > 0, \cr & v_t = \Delta v-v + u,\quad x\in \Omega ,\;t > 0.} $$
It is shown that whenever ρ ∈ ℝ, μ > 0 and
$$l > \max \left\{ {\displaystyle{{n + 2} \over 2},2} \right\},$$
then the considered system possesses a global classical solution for all sufficiently smooth initial data. Furthermore, the solution converges to the equilibrium
$$\left( {{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)},{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)}} \right)$$
as t → ∞ under some extra hypotheses, where ρ+ = max{ρ, 0}.

Keywords

Global existenceBoundednessLarge time behaviourSignal-dependent motility

MSC classification

Primary: 35A01: Existence problems
Secondary: 35B45: A priori estimates 35K51: Initial-boundary value problems for second-order parabolic systems 92C17: Cell movement (chemotaxis, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 821 - 841
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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