Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 77, pp. 1-17.

Title: Existence of global solutions for cross-diffusion models in a fractional setting
       
Authors: Jhean E. Perez-Lopez (Univ. Industrial de Santander, Bucaramanga, Colombia)
         Diego A. Rueda-Gomez (Univ. Industrial de Santander, Bucaramanga, Colombia)
         Elder J. Villamizar-Roa (Univ. Industrial de Santander, Bucaramanga, Colombia)
         
Abstract:
This article is devoted to the analysis of a fractional chemotaxis model in
R^N with a time fractional variation in the Caputo sense and a
fractional spatial diffusion. This model encompasses the fractional Keller-Segel
system [9] which describes the movement of living organisms towards higher
concentration regions of chemical attractants, and a fractional Lotka-Volterra
competition model [16] describing the competition interspecies 
in which one of the competing species avoids encounters with rivals by 
means of chemorepulsion. We prove product estimates in Besov-Morrey spaces 
and derive global estimates for mild solutions of the fractional heat equation. 
We use these results to prove the existence and uniqueness of global mild solutions for the differential system in a framework of Besov-Morrey spaces.

Submitted August 6, 2022. Published November 10, 2023.
Math Subject Classifications: 35K55, 35Q35, 35Q92, 92C17.
Key Words: Keller-Segel system; Lotka-Volterra; cross-diffusion; Besov-Morrey spaces.