Infinite time blow-up of solutions to the classical Keller-Segel model of chemotaxis in higher dimensions

We study the simplest parabolic-elliptic model of chemotaxis in ${\mathbf{R}}^{\mathbf{N}}$, and show the optimal criteria for the existence of infinite time blow-up of solutions with the initial data below the Chandrasekhar singular solution when $N\ge 10$. In particular, we exhibit the differences of the criteria between the cases $N\ge 11$ and $N=10$. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution and the asymptotic expansions of solutions to the Liouville equation.