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

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关于半线性抛物方程组解的解析半径

我们研究了尺度不变半线性抛物方程组的强解的空间解析半径$ R（t）$。众所周知，在初始时间附近，$ R（t）t ^ {-\ frac {1} {2}} $由一个正常数从下方限制。在本文中，我们证明$ \ displaystyle \ liminf_ {t \ to 0} R（t）t ^ {-\ frac {1} {2}} = \ infty $，并假设初始数据具有较高的规律性，我们可以获得零时间附近的改进下限。作为应用程序，我们证明对于$ C（[0，\ infty）; H ^ {\ frac {1} {2}}（\ mathbb {R} ^ 3））$的Navier–Stokes方程，其中有$ \ displaystyle \ liminf_ {t \ to \ infty} R（t）t ^ {-\ frac {1} {2}} = \ infty $。