The rate of concentration for the radially symmetric solution to a degenerate drift-diffusion equation with the mass critical exponent

We consider the concentration rate of the total mass for radially symmetric blow-up solutions to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. We proved that the radially symmetric solution blows up in finite time when the initial data has negative free energy. We show that the mass concentration phenomenon occurs with the sharp lower constant related to the best constant of the Hardy-Littlewood-Sobolev inequality and the concentration rate of the total mass.