Sharp pointwise estimates for solutions of weakly coupled second order parabolic system in a layer

We deal with $m$-component vector-valued solutions to the Cauchy problem for linear both homogeneous and nonhomogeneous weakly coupled second order parabolic system in the layer ${\mathbb{R}}_T^{n+1}={\mathbb{R}}^n\times (0,T)$. We assume that coefficients of the system are real and depending only on $t$, $n\ge 1$ and $T<\mathrm{\infty }$. The homogeneous system is considered with initial data in $[L^p({\mathbb{R}}^n){]}^m$, $1\le p\le \mathrm{\infty }$. For the nonhomogeneous system we suppose that the initial function is equal to zero and the right-hand side belongs to $[L^p({\mathbb R}^{n+1}_T)]^m\cap [C^\alpha \big (\overline{{\mathbb R}^{n+1}_T} \big )]^m $,$\alpha \in (0, 1)$. Explicit formulas for the sharp coefficients in pointwise estimates for solutions of these problems and their directional derivative are obtained.