Single-point blow-up in the Cauchy problem for the higher-dimensional Keller-Segel system

The Cauchy problem in R-n for the Keller-Segel system {u(t) = Delta u - del center dot (u del v), v(t) = Delta v - v + u, is considered for n >= 3. Using a basic theory of local existence and maximal extensibility of classical and spatially integrable solutions as a starting point, the study provides a result on the occurrence of finite-time blow-up within considerably large sets of radially symmetric initial data, and moreover verifies that any such explosion exclusively occurs at the spatial origin. The detection of blow-up is accomplished by analyzing a relative of the well-known Keller-Segel energy inequality, involving a modification of the corresponding energy functional which, unlike the latter, can be seen to be favourably controlled from below by the corresponding dissipation rate through a certain functional inequality along trajectories.