Nondegeneracy of blow-up points for the parabolic Keller-Segel system

This paper is concerned with the parabolic Keller-Segel system {u(t) = del center dot (del u - u(m)del v) in Omega x (0, T), Gamma v(t) - Delta v - lambda v + u in Omega x (0, T), in a domain Omega of R-N with N >= 1, where m, Gamma > 0, lambda >= 0 are constants and T > 0. When Omega not equal R-N, we impose the Neumann boundary conditions on the boundary. Under suitable assumptions, we prove the local nondegeneracy of blow-up points. This seems new even for the classical Keller Segel system (m = 1). Lower global blow-up estimates are also obtained. In the singular case 0 <m < 1, as a prerequisite, local existence and regularity properties are established. (C) 2013 Elsevier Masson SAS. All rights reserved.