On weakly coupled systems of partial differential equations with different diffusion terms

We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space $C_b(\mathbb{R}^d;\mathbb{R}^m)$ of bounded and continuous functions, associated to a class of nonautonomous weakly coupled second-order elliptic operators $\bf{\mathcal A}$, with possibly unbounded coefficients and diffusion and drift terms which vary from equation to equation. We also provide estimates of the spatial derivatives up to the third-order and continuity properties both of the evolution operator ${\bf G}(t,s)$ associated to the Cauchy problem $D_t{\bf u}=\bf{\mathcal A}(t){\bf u}$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$, and, for fixed $\overline t$, of the semigroup ${\bf T}_{\overline t}(\tau)$ associated to the autonomous Cauchy problem $D_{\tau}{\bf u}={\bf{\mathcal A}}(\overline t){\bf u}$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$. These results allow us to deal with elliptic problems whose coefficients also depend on time.