Hölder regularity and uniqueness theorem on weak solutions to the degenerate Keller–Segel system. (June 2016)

• Record Type: Journal Article
• Authors: Kim, Sunghoon; Lee, Ki-Ahm
• Is Part Of: Nonlinear analysis. Volume 138(2016)
• Page Start: 229
• Page End: 252
• Publication Date: 2016-06
• Subjects: Degenerate Keller–Segel system -- Hölder estimate -- Uniqueness; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.11.024
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we present local Hölder 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 Hölder continuous functions by proving L 1 -contraction in this class.