The aggregation-diffusion equation with energy critical exponent

Abstract: We consider a Keller-Segel model with non-linear porous medium type diffusion and nonlocal attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen to be $m=\frac{2d}{d+2s}$ in such a way that the associated free energy is conformal invariant and there is a family of stationary solutions $U(x)=c\left(\frac{\lambda}{\lambda^2+|x-x_0|^2}\right)^{\frac{d+2s}{2}}$ for any constant $c$ and some $\lambda>0, x_0 \in \R^d.$ We analyze under which conditions on the initial data the regime that attractive forces are stronger than diffusion occurs and classify the global existence and finite time blow-up of dynamical solutions by virtue of stationary solutions. Precisely, solutions exist globally in time if the $L^m$ norm of the initial data $|u_0|{L^m(\R^d)}$ is less than the $L^m$ norm of stationary solutions $|U(x)|{L^m(\R^d)}$. Whereas there are blowing-up solutions for $|u_0|{L^m(\R^d)}>|U(x)|{L^m(\R^d)}$.