Long-time dynamics of classical Patlak-Keller-Segel equation

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 $p \in [1,\infty]$.

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