On the radius of analyticity of solutions to semi-linear parabolic systems

Abstract

We study the radius of analyticity~$R(t)$ in space, of strong solutions to systems of scale-invariant semi-linear parabolic equations. It is well-known that near the initial time,~$R(t)t^{-\frac{1}{2}}$ is bounded from below by a positive constant. In this paper we prove that~$\underset{t\to 0}{\mathrm{liminf}}R(t)t^{-\frac{1}{2}}=\infty$, and assuming higher regularity for the initial data, we obtain an improved lower bound near time zero. As an application, we prove that for any global solution~$u\in C([0,\infty );H^{\frac{1}{2}}({\mathbb{R}}^3))$ of the Navier-Stokes equations, there holds~$\underset{t\to \infty }{\mathrm{liminf}}R(t)t^{-\frac{1}{2}}=\infty$.