  When the spatial dimension $n =2$, it has been well-known that a global mild
solution to classical Patlak-Keller-Segel equation (PKS equation for short)
exists if and only if its initial total mass is not in supercritical regime.
However, to study long-time behavior of a global mild solution to $2$D PKS
equation usually requires finite-free-energy and finite-second-moment
assumptions on initial data. In this article, we introduce a novel argument to
push and stretch a space-time strip. By this way, we gain $L^1$-compactness of
PKS equation expressed under similarity variables. As a consequence, we obtain
global dynamics of 2D PKS equation in subcritical regime with no additional
assumptions. As for the higher dimensional case in which the spatial dimension
$n \geq 3$, we also characterize the long-time asymptotics of global mild
solutions to PKS equation. With a finite-total-mass assumption on density of
cells, any global mild solution to PKS equation will approach a self-similar
profile when time is large, provided that there is a sequence of time going to
infinity on which $L^\infty$-norm of the density of cells converges to zero.
The self-similar profile in the higher dimensional case is given by the
function $M\mathcal{G}_n$, where $M$ is the total mass of cells and
$\mathcal{G}_n$ denotes the standard $n$-dimensional Gaussian probability
density. Convergence rates to self-similar profiles are also discussed in any
dimensions. Particularly in the higher dimensional case, the general
convergence rate for $L^1$-initial data can be improved if the initial data has
a finite second moment. In fact, when time is large and $n \geq 3$, we provide,
in an optimal way, a higher-order approximation of global mild solutions to PKS
equation if the initial density has a finite second moment. All convergence
rates studied in this article are under the $L^p$-norm with $p \in [1,\infty]$.

Analysis of PDEs