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 ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$. We assume that coefficients of the system are real and depending only on $t$, $n\geq 1$ and $T<\infty$. The homogeneous system is considered with initial data in $[L^p({\mathbb R}^n)]^m$, $1\leq p \leq \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.

中文翻译：

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层中弱耦合二阶抛物线系统解的逐点估计

我们处理 ${\mathbb R}^{n+1}_T={\mathbb R} 层中线性齐次和非齐次弱耦合二阶抛物线系统的柯西问题的 $m$-分量向量值解^n\times (0, T)$。我们假设系统的系数是实数，并且仅取决于 $t$、$n\geq 1$ 和 $T<\infty$。齐次系统被认为具有 $[L^p({\mathbb R}^n)]^m$, $1\leq p \leq \infty $ 中的初始数据。对于非齐次系统，我们假设初始函数为零，右侧属于 $[L^p({\mathbb R}^{n+1}_T)]^m\cap [C^\ alpha \big (\overline{{\mathbb R}^{n+1}_T} \big )]^m $, $\alpha \in (0, 1)$。获得了这些问题的解决方案的逐点估计中尖锐系数的显式公式及其方向导数。