We study the Cauchy problem associated to parabolic systems of the form $D_t\boldsymbol{u}=\boldsymbol{\mathcal A}(t)\boldsymbol u$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$, the space of continuous and bounded functions $\boldsymbol{f}:\mathbb{R}^d\to\mathbb{R}^m$. Here $\boldsymbol{\mathcal A}(t)$ is a weakly coupled elliptic operator acting on vector-valued functions, having diffusion and drift coefficients which change from equation to equation. We prove existence and uniqueness of the evolution operator $\boldsymbol{G}(t,s)$ which governs the problem in $C_b(\mathbb{R}^d;\mathbb{R}^m)$ proving its positivity. The compactness of $\boldsymbol{G}(t,s)$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$ and some of its consequences are also studied. Finally, we extend the evolution operator $\boldsymbol{G}(t,s)$ to the $L^p$- spaces related to the so called "evolution system of measures" and we provide conditions for the compactness of $\boldsymbol{G}(t,s)$ in this setting.

中文翻译：

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具有无界系数的偏微分方程耦合系统

我们研究与 $D_t\boldsymbol{u}=\boldsymbol{\mathcal A}(t)\boldsymbol u$ 形式的抛物线系统相关的柯西问题，在 $C_b(\mathbb{R}^d;\mathbb{R }^m)$，连续有界函数空间$\boldsymbol{f}:\mathbb{R}^d\to\mathbb{R}^m$。这里 $\boldsymbol{\mathcal A}(t)$ 是一个弱耦合椭圆算子，作用于向量值函数，具有从方程到方程变化的扩散和漂移系数。我们证明了控制 $C_b(\mathbb{R}^d;\mathbb{R}^m)$ 中问题的演化算子 $\boldsymbol{G}(t,s)$ 的存在性和唯一性，证明了它的正性。还研究了 $\boldsymbol{G}(t,s)$ 在 $C_b(\mathbb{R}^d;\mathbb{R}^m)$ 中的紧凑性及其一些后果。最后，我们将演化算子 $\boldsymbol{G}(t,s)$ 扩展到与所谓的“