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:\\ \[ \begin{aligned} &\mbox{there is no global solution for any initial data if and only if }\\ &\mbox{the function } f \mbox{ satisfies}\\ &\hspace{20mm}\int_{0}^{\infty}\psi(t)\frac{f\left(\epsilon \,\left\| S(t)u_{0}\right\|_{\infty}\right)}{\left\| S(t)u_{0}\right\|_{\infty}}dt=\infty\\ &\mbox{for every }\,\epsilon>0\,\mbox{ and nonnegative nontrivial initial data }\,u_{0}\in C_{0}(\Omega). \end{aligned} \] Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the Dirichlet boundary condition.