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

TL;DR

This paper investigates the long-time behavior of solutions to the classical Patlak-Keller-Segel equation across different dimensions, establishing new methods to analyze global dynamics without restrictive initial data assumptions.

Contribution

Introduces a novel argument to analyze 2D PKS equation without finite-energy assumptions and characterizes long-time asymptotics for higher dimensions with improved convergence rates.

Findings

Global dynamics in 2D without additional assumptions.

Solutions approach Gaussian profiles in higher dimensions.

Enhanced convergence rates with finite second moment initial data.

Abstract

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