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# Mathematical Research Letters

## Volume 27 (2020)

### Number 6

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

Pages: 1631 – 1643

DOI: https://dx.doi.org/10.4310/MRL.2020.v27.n6.a2

#### Authors

#### 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 $\displaystyle\liminf_{t\to 0} R(t)t^{-\frac{1}{2}}=\infty$, 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 $\displaystyle\liminf_{t\to \infty} R(t)t^{-\frac{1}{2}}= \infty$.

Part of the work was done when P. Zhang was visiting Laboratoire J. L. Lions of Sorbonne Université in the fall of 2018. He would like to thank the hospitality of the Laboratory. P. Zhang is partially supported by NSF of China under Grants 11731007 and 11688101, and innovation grant from National Center for Mathematics and Interdisciplinary Sciences. This work is also supported by the K.C. Wong Education Foundation.

Received 8 April 2020

Accepted 13 October 2020

Published 17 February 2021