Finite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition

We consider the non-existence and the non-uniform boundedness of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. If the initial data has negative free energy, then either the corresponding weak solution to the equation does not exist globally in time, or the time global solution does not remain bounded in the energy space. We emphasize that our result does not require any weight assumption on the initial data, and hence, a solution may have an infinite second moment. The proof is based upon the modified virial law and conservation laws and we show that the modified moment functional vanishes for a finite time under the negative energy condition. For a radially symmetric case, the solution blows up in finite time and the mass concentration phenomenon occurs with a sharp lower bound related to the best constant for the Hardy–Littlewood–Sobolev inequality.