Yamabe flow on non-compact manifolds with unbounded initial curvature

We prove global existence of Yamabe flows on non-compact manifolds $M$ of dimension $m\ge 3$ under the assumption that the initial metric $g_0=u_0{g}_M$ is conformally equivalent to a complete background metric $g_M$ of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor $u_0$ bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of $(M,g_0)$ can be unbounded from above and below without growth condition.