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

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