Dirichlet conditions at infinity for parabolic and elliptic equations

We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in RN×R+RN×R+. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is addressed. We consider also elliptic equations in RNRN with similar conditions at infinity.