Generalized Doubly Parabolic Keller-Segel System with Fractional Diffusion

TL;DR

This paper investigates a generalized Keller-Segel model with fractional diffusion in multiple dimensions, establishing well-posedness and asymptotic behavior, thus extending classical chemotaxis models to include superdiffusive effects.

Contribution

It introduces a doubly fractional Keller-Segel system with distinct exponents, proving local and global well-posedness, and analyzing long-term dynamics, which is a novel extension of classical models.

Findings

Established local well-posedness of solutions.

Proved global well-posedness under small initial data.

Characterized the asymptotic behavior of solutions.

Abstract

The Keller-Segel model is a system of partial differential equations that describes the movement of cells or organisms in response to chemical signals, a phenomenon known as chemotaxis. In this study, we analyze a doubly parabolic Keller-Segel system in the whole space $\mathbb{R}^d$, $d\geq 2$, where both cellular and chemical diffusion are governed by fractional Laplacians with distinct exponents. This system generalizes the classical Keller-Segel model by introducing superdiffusion, a form of anomalous diffusion. This extension accounts for nonlocal diffusive effects observed in experimental settings, particularly in environments with sparse targets. We establish results on the local well-posedness of mild solutions for this generalized system and global well-posedness under smallness assumptions on the initial conditions in $L^p(\mathbb{R}^d)$. Furthermore, we characterize the asymptotic behavior of the solution.