Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

Jan Burczak; Rafael Granero-Belinchón

doi:10.3934/dcdss.2020008

Article Contents

2020, Volume 13, Issue 2: 139-164. Doi: 10.3934/dcdss.2020008

This issue Previous Article Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity Next Article Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity

Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

Jan Burczak^1,2, and
Rafael Granero-Belinchón^3,

1.
Institute of Mathematics, Polish Academy of Sciences, Warsaw, Śniadeckich 8, 00-656, Poland
2.
OxPDE, Mathematical Institute, University of Oxford, UK
3.
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n, Santander, Spain

Received: April 30, 2017

Published: February 2020

JB was supported by the National Science Centre, Poland (NCN) grant 'SONATA' 2016/21/D/ST1/03085. RGB was partially supported by the Grant MTM2014-59488-P from the Ministerio de Economía y Competitividad (MINECO, Spain).

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Abstract

In this paper we consider a $ d $-dimensional ($ d = 1, 2 $) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order $ \alpha \in (0, 2) $. We prove uniform in time boundedness of its solution in the supercritical range $ \alpha>d\left(1-c\right) $, where $ c $ is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for $ \|u(t)-u_\infty\|_{L^\infty}\rightarrow0 $, where $ u_\infty\equiv 1 $ is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.

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Mathematics Subject Classification: 35B65, 35R11, 92C17, 35A01, 35S10.

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