H\"older regularity and Uniqueness theorem on weak solutions to the degenerate Keller-Segel system

TL;DR

This paper establishes local Hölder regularity and uniqueness of weak solutions for the degenerate Keller-Segel system in certain parameter ranges, extending understanding of solution behavior before blow-up occurs.

Contribution

It provides the first Hölder estimates for the degenerate Keller-Segel system and proves uniqueness of weak solutions within Hölder continuous functions.

Findings

Established local Hölder regularity for solutions.

Proved uniqueness of weak solutions in Hölder class.

Developed uniform estimates based on density sup-norm.

Abstract

In this paper, we present local H\"older estimates for the degenerate Keller-Segel system \eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} below in the range of $m>1$ and $q>1$ before a blow-up of solutions. To deal with difficulties caused by the degeneracy of the operator, we find uniform estimates depending sup-norm of the density function and modified the energy estimates and intrinsic scales considered in Porous Medium Equation. As its application, the uniqueness of weak solution to \eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} is also showed in the class of H\"older continuous functions by proving $L^1$-contraction in this class.