Stability and instability for the fully parabolic Keller-Segel system around constant equilibrium

This paper studies the Cauchy problem for the fully parabolic Keller-Segel system. The main results show that there exists a critical threshold $A_{\rm crit}>0$ for steady states $(A,A)$ such that the steady states are nonlinearly stable when $A\le A_{\rm crit}$ and nonlinearly unstable when $A>A_{\rm crit}$. We discuss asymptotic convergence rates as well. In the subcritical case $A<A_{\rm crit}$, the rates correspond to those of the heat equation, and in the critical case $A=A_{\rm crit}$, the rates correspond to half those of the heat equation.