Quantitative analysis and its applications for Keller-Segel type systems

TL;DR

This paper employs De Giorgi iteration to derive quantitative upper bounds for Keller-Segel systems, enabling insights into long-term behavior, regularity, and stability of solutions in chemotaxis models involving nonlinear diffusion and fluid interaction.

Contribution

It introduces a refined upper bound estimate for Keller-Segel systems and applies it to analyze stability and regularity in various chemotaxis models with nonlinear and fluid dynamics components.

Findings

Established asymptotic stability of chemotaxis models.

Proved Hölder continuity for models with p-Laplacian diffusion.

Improved regularity results for chemotaxis-haptotaxis and chemotaxis-Navier-Stokes models.

Abstract

In this paper, we utilize the De Giorgi iteration to quantitatively analyze the upper bound of solutions for Keller-Segel type systems. The refined upper bound estimate presented here has broad applications in determining large time behaviours of weak solutions and improving the regularity for models involving the $p$-Laplace operator. To demonstrate the applicability of our findings, we investigate the asymptotic stability of a chemotaxis model with nonlinear signal production and a chemotaxis-Navier-Stokes model with a logistic source. Additionally, within the context of $p$-Laplacian diffusion, we establish H\"{o}lder continuity for a chemotaxis-haptotaxis model and a chemotaxis-Stokes model.