  We investigate existence and uniqueness of bounded solutions of parabolic
equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a
complete noncompact Riemannian manifold. 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 studied. We consider also elliptic
equations on $M$ with similar conditions at infinity.
