Finite-time blow-up for solutions to a degenerate drift-diffusion equation for a fast-diffusion case

We consider the non-existence of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with a fast-diffusion exponent. We show that the solution for fast-diffusion cases with the diffusion exponent n/n+2 < alpha < 1 blows up in a finite time if the initial data satisfy certain conditions involving the free energy. We also show the finite-time blow-up for the radially symmetric case without a finite moment condition.