  We deal with solutions of the Cauchy problem to linear both homogeneous and
nonhomogeneous parabolic second order equations with real constant coefficients
in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$, where $n\geq 1$
and $T<\infty$. The homogeneous equation is considered with initial data in
$L^p({\mathbb R}^n)$, $1\leq p \leq \infty $. For the nonhomogeneous equation
we suppose that initial function is equal to zero and the function in the
right-hand side belongs to $f\in L^p({\mathbb R}^{n+1}_T)\cap C^\alpha \big
(\bar{{\mathbb R}^{n+1}_T} \big ) $ , $p>n+2$ and $\alpha \in (0, 1)$. Explicit
formulas for the sharp coefficients in pointwise estimates for the length of
the gradient to solutions to these problems are obtained.
