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arXiv · 1707.04527

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

Jan Burczak·Rafael Granero-Belinchón
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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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Jan Burczak, Rafael Granero-Belinchón. 2017-07-14. Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations. https://arxiv.org/abs/1707.04527

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