  In this paper, we give a clear cut relation between the the volume growth
$V(r)$ and the existence of nonnegative solutions to parabolic semilinear
problem \begin{align}\tag{*}\label{*} \left\{ \begin{array}{ll} \Delta u -
\partial_t u + u^p = 0, \\ {u(x,0)= {u_0(x)}}, \end{array} \right. \end{align}
on a large class of Riemannian manifolds. We prove that for parameter $p>1$, if
\begin{align*} \int^{+\infty} \frac{t}{V(t)^{p-1}} dt = \infty \end{align*}
then (\ref{*}) has no nonnegative solution. If \begin{align*} \int^{+\infty}
\frac{t}{V(t)^{p-1}} dt < \infty \end{align*} then (\ref{*}) has positive
solutions for small $u_0$.
