Hölder regularity and uniqueness theorem on weak solutions to the degenerate Keller–Segel system

In this paper, we present local Hölder estimates for the degenerate Keller–Segel system (KSm)(KSm) below in the range of m>1m>1 and q>2q>2 before a blow-up of solutions. To deal with difficulties caused by the degeneracy of the operator, we find uniform estimates depending on 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 (KSm)(KSm) is also showed for the case q>max(1+m2,2) in the class of Hölder continuous functions by proving L1L1-contraction in this class.