  The purpose of this paper is to give a necessary and sufficient condition for
the existence and non-existence of global solutions of the following semilinear
parabolic equations
  \[
  u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times (0,t^{*}),
  \]
  under the Dirichlet boundary condition on a bounded domain. In fact, this has
remained as an open problem for a few decades, even for the case $f(u)=u^{p}$.
As a matter of fact, we prove:
  \[
  \begin{aligned}
  &\mbox{there is no global solution for any initial data if and only if }
  &\mbox{the function } f \mbox{ satisfies}
  &\hspace{20mm}\int_{0}^{\infty}\psi(t)\frac{f\left(\lVert
S(t)u_{0}\rVert_{\infty}\right)}{\lVert S(t)u_{0}\rVert_{\infty}}dt=\infty
  &\mbox{for every }\,\epsilon>0\,\mbox{ and nonnegative nontrivial initial
data }\,u_{0}\in C_{0}(\Omega).
  \end{aligned}
  \]
  Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the Dirichlet boundary
condition.
