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

TL;DR

This paper investigates a fractional Keller-Segel model with logistic growth in one and two dimensions, establishing boundedness, convergence to equilibrium, and uniqueness of solutions under certain conditions.

Contribution

It introduces new conditions for boundedness and convergence in a fractional Keller-Segel model with logistic forcing, extending understanding of its long-term behavior.

Findings

Solutions are uniformly bounded in time for certain fractional orders.

Conditions are identified under which solutions converge to the homogeneous steady state.

A uniqueness result for solutions is established.

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.