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 $A_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}>0$ for steady states $(A,A)$ such that the steady states are nonlinearly stable when $A\le A_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ and nonlinearly unstable when $A>A_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$. We discuss asymptotic convergence rates as well. In the subcritical case $A<A_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$, the rates correspond to those of the heat equation, and in the critical case $A=A_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$, the rates correspond to half those of the heat equation.