Generalized Doubly Parabolic Keller-Segel System with Fractional Diffusion

2025/01/16 by Bronzi, Anne Caroline, de Andrade, Crystianne Lilian
#35K55 #35Q92 #92B99 #92C17 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.09871

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 ℝ^d, d≥ 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(ℝ^d). Furthermore, we characterize the asymptotic behavior of the solution.

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