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 Cb(Rd;Rm) of bounded and continuous functions, associated to a class of nonautonomous weakly coupled secondorder elliptic operators 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 G(t, s) associated to the Cauchy problem Dtu = A(t)u in Cb(Rd;Rm), and, for fixed t, of the semigroup Tt(τ) associated to the autonomous Cauchy problem Dτu = A(t)u in Cb(Rd;Rm). These results allow us to deal with elliptic problems whose coefficients also depend on time.