On coupled systems of PDEs with unbounded coefficients

We study the Cauchy problem associated with parabolic systems of the form Dtu = A(t)u in Cb (Rd; Rm), the space of continuous and bounded functions f: Rd → Rm. Here A(t) is a coupled nonautonomous 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 G(t, s) which governs the problem in Cb (Rd; Rm) and its positivity. The compactness of G(t, s) inCb (Rd; Rm) and some of its consequences are also studied. Finally, we extend the evolution operator G(t, s) to the Lp-spaces related to the so called “evolution system of measures” and we provide conditions for the compactness of G(t, s) in this setting.