Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity

Tomasz Cieślak; Kentarou Fujie

doi:10.3934/dcdss.2020009

Article Contents

2020, Volume 13, Issue 2: 165-176. Doi: 10.3934/dcdss.2020009

This issue Previous Article Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations Next Article On a chemotaxis model with competitive terms arising in angiogenesis

Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity

Tomasz Cieślak^1, , and
Kentarou Fujie^2,

1.
Institute of Mathematics, Polish Academy of Sciences, Warsaw, 00-656, Poland
2.
Department of Mathematics, Tokyo University of Science, Tokyo, 162-8601, Japan

^* Corresponding author: Tomasz Cieślak
^* Corresponding author: Tomasz Cieślak

Received: May 2017

Revised: January 2018

Published: February 2020

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Abstract

The paper should be viewed as complement of an earlier result in [10]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear diffusion of the form $ 1/u $ all the solutions, independently on the magnitude of initial mass, stay bounded. However, the argument presented in [10] deals with the Jäger-Luckhaus type system. And is very sensitive to this restriction. Namely, the change of variables introduced in [10], being a main step of the method, works only for the Jäger-Luckhaus modification. It does not seem to be applicable in the usual version of the parabolic-elliptic Keller-Segel system. The present paper fulfils this gap and deals with the case of the usual parabolic-elliptic version. To handle it we establish a new Lyapunov-like functional (it is related to what was done in [10]), which leads to global existence of the initial-boundary value problem for any initial mass.

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Mathematics Subject Classification: Primary: 35B45, 35K45; Secondary: 35Q92, 92C17.

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References

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