Generalized Doubly Parabolic Keller-Segel System with Fractional Diffusion
Anne Caroline Bronzi, Crystianne Lilian de Andrade
TL;DR
The paper addresses the generalized doubly parabolic Keller–Segel system with fractional diffusion in $\mathbb{R}^d$, establishing local and global well-posedness for mild solutions under precise Lebesgue-space conditions on the initial data and using a fixed-point framework based on Duhamel representations with the kernels $K_t^{\alpha}$ and $K_t^{\beta}$. Local existence is obtained in a time interval $[0,T]$ via a contraction mapping in a suitable Banach space, while global existence is achieved under a smallness condition on the initial data, accompanied by decay estimates and mass conservation. The analysis also provides higher-regularity results in Sobolev settings and guarantees nonnegativity of solutions when starting from nonnegative data. Collectively, the results extend classical KS theory to a fractional, parabolic-parabolic regime, quantify diffusion-advection interactions, and illuminate long-time behavior and stability under fractional diffusion.
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.