  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,~$R(t)t^{-\frac12}$ is bounded from below by a
positive constant. In this paper we prove
that~$\displaystyle\liminf_{t\rightarrow 0} R(t)t^{-\frac12}= \infty$, 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~$u\in C([0,\infty); H^{\frac12}(\R^3))$ of the Navier-Stokes
equations, there holds~$\displaystyle\liminf_{t\rightarrow \infty}
R(t)t^{-\frac12}= \infty$.
