On the blow-up profile of Keller-Segel-Patlak system

In this paper, we obtain the sharp estimate on the asymptotic behaviors of blow-up profiles for Keller-Segel-Patlak system in the space with dimensions N >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document}: 0.1ut=Delta u-del<middle dot>(u del v),x is an element of RN,t is an element of(0,T),0=Delta v+u,x is an element of RN,t is an element of(0,T),u(x,0)=u0(x),x is an element of RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u\nabla v), & x\in \mathbb {R}<^>N, \ t\in (0,T), \\ 0 = \Delta v+u, & x\in \mathbb {R}<^>N, \ t\in (0,T), \\ u(x,0)=u_0(x), & x\in \mathbb {R}<^>N, \end{array} \right. \end{aligned}$$\end{document}which solves an open problem proposed by Souplet and Winkler in [41]. To establish this result, we develop the zero number argument for nonlinear equations with unbounded coefficients and construct a family of auxiliary backward self-similar solutions through nontrivial ODE analysis.