On the Cauchy problem for the fractional Keller-Segel system in variable Lebesgue spaces

TL;DR

This paper investigates the well-posedness of the fractional Keller-Segel system within variable Lebesgue spaces, transforming the system into a nonlinear heat equation and leveraging decay estimates to establish key results.

Contribution

It introduces a novel approach by reducing the fractional Keller-Segel system to a generalized heat equation in variable Lebesgue spaces, overcoming boundedness challenges.

Findings

Established well-posedness results for the fractional Keller-Segel system in variable Lebesgue spaces.

Developed a reduction technique to a nonlinear heat equation to handle boundedness issues.

Utilized decay estimates of the fractional heat kernel to prove key properties.

Abstract

In this paper, we are mainly concerned with the well-posed problem of the fractional Keller--Segel system in the framework of variable Lebesgue spaces. Based on carefully examining the algebraical structure of the system, we reduced the fractional Keller--Segel system into the generalized nonlinear heat equation to overcome the difficulties caused by the boundedness of the Riesz potential in a variable Lebesgue spaces, then by mixing some structural properties of the variable Lebesgue spaces with the optimal decay estimates of the fractional heat kernel, we were able to establish two well-posedness results of the fractional Keller--Segel system in this functional setting.