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Given a class of nonautonomous elliptic operators $\A(t)$ with unboundedcoefficients, defined in $\overline{I \times \Om}$ (where $I$ is aright-halfline or $I=\R$ and $\Om\subset \Rd$ is possibly unbounded), we proveexistence and uniqueness of the evolution operator associated to $\A(t)$ in thespace of bounded and continuous functions, under Dirichlet and first order, nontangential homogeneous boundary conditions. Some qualitative properties of thesolutions, the compactness of the evolution operator and some uniform gradientestimates are then proved.

Non autonomous parabolic problems with unbounded coefficients in unbounded domains