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

Abstract In this paper, we present local Holder estimates for the degenerate Keller–Segel system ( K S m ) below in the range of m > 1 and q > 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 ( K S m ) is also showed for the case q > max ( 1 + m 2 , 2 ) in the class of Holder continuous functions by proving L 1 -contraction in this class.