The aggregation-diffusion equation with energy critical exponent

TL;DR

This paper studies a Keller-Segel model with energy-critical nonlinear diffusion and nonlocal attraction, classifying conditions for global existence versus finite-time blow-up based on initial data relative to stationary solutions.

Contribution

It introduces a conformally invariant Keller-Segel model with explicit stationary solutions and characterizes the initial data thresholds for global existence and blow-up.

Findings

Solutions exist globally if initial data norm is below stationary solution norm.

Solutions blow up in finite time if initial data norm exceeds stationary solution norm.

Stationary solutions are explicitly characterized and used as thresholds.

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 $\|u_0\|_{L^m(\R^d)}>\|U(x)\|_{L^m(\R^d)}$.