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

Abstract

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 $ {\mathit{\boldsymbol{\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 $ {{\mathit{\boldsymbol{G}}}}(t, s) $ associated to the Cauchy problem $ D_t {{\mathit{\boldsymbol{u}}}} = {\mathit{\boldsymbol{\mathcal A}}}(t) {{\mathit{\boldsymbol{u}}}} $ in $ C_b( \mathbb R^d; \mathbb R^m) $, and, for fixed $ \overline t $, of the semigroup $ {{\mathit{\boldsymbol{T}}}}_{\overline t}(\tau) $ associated to the autonomous Cauchy problem $ D_\tau {{\mathit{\boldsymbol{u}}}} = {\mathit{\boldsymbol{\mathcal A}}}(\overline t) {{\mathit{\boldsymbol{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.