Optimal decay rates and space–time analyticity of solutions to the Patlak-Keller–Segel equations

Abstract

Based on some new elementary estimates for the space–time derivatives of the heat kernel, we use a bootstrapping approach to establish quantitative estimates on the optimal decay rates for the $L^q\left(R^d\right)$ ($1\le q\le \infty$, $d\in N$) norm of the space–time derivatives of solutions to the (modified) Patlak-Keller–Segel equations with initial data in $L^1\left(R^d\right)$, which implies the joint space–time analyticity of solutions. When the $L^1\left(R^d\right)$ norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the radius of space–time analyticity in time. As a byproduct, the space analyticity is obtained for any initial data in $L^1\left(R^d\right)$. The decay estimates and space–time analyticity are also established for solutions bounded in both space and time variables. The results can be extended to a more general class of equations.