Non-existence of finite-time self-similar solutions of the Keller–Segel system in the scaling invariant class

Let us consider the problem whether there does exist a finite-time self-similar solution of the backward type to the semilinear Keller–Segel system. In the case of parabolic–elliptic type for n ⩾ 3 we show that there is no such a solution with a finite mass in the scaling invariant class. On the other hand, in the case of parabolic–parabolic type for n ⩾ 2 , non-existence of finite-time self-similar solutions is proved in a larger class of a finite mass with some local bounds.