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Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity

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

 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  May 2017 Published  January 2019

Fund Project: 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)

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.

Citation: Jan Burczak, Rafael Granero-Belinchón. Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020008
References:
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References:
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