  In this paper, we study the existence of global Yamabe flow on asymptotically
flat (in short, AF or ALE) manifolds. Note that the ADM mass is preserved in
dimensions 3,4 and 5. We present a new general local existence of Yamabe flow
on a complete Riemannian manifold with the initial metric quasi-isometric to a
background metric of bounded scalar curvature. Asymptotic behaviour of the
Yamabe flow on ALE manifolds is also addressed provided the initial scalar
curvature is non-negative and there is a bounded subsolution to the
corresponding Poisson equation. We also present a maximum principle for a very
general parabolic equations on the complete Riemannian manifolds.
