  We prove the analyticity in time for solutions of two parabolic equations in
the whole space, without any decaying or vanishing conditions. One of them
involves solutions to the heat equation of exponential growth of order $2$ on
$\M$. Here $\M$ is $\R^d$ or a complete noncompact manifold with Ricci
curvature bounded from below by a constant. An implication is a sharp
solvability condition for the Cauchy problem of the backward heat equation,
which is a well known ill-posed problem. Another implication is a sharp
criteria for time analyticity of solutions down to the initial time. The other
pertains bounded mild solutions of the incompressible Navier-Stokes equations
in the whole space.
  There are many long established analyticity results for the Navier-Stokes
equations. See for example \cite{Ka:1} and \cite{FT:1} for spatial and time
analyticity in certain integral sense, \cite{CN:1} for pointwise space-time
analyticity of 3 dimensional solutions to the Cauchy problem, and also the
pointwise time analyticity results of \cite{Ma:1} and \cite{Gi:1} under zero
boundary condition. Our result seems to be the first general pointwise time
analyticity result for the Cauchy problem for all dimensions, whose proof
involves only real variable method. The proof involves a method of
algebraically manipulating the integral kernel, which appears applicable to
other evolution equations.
