  We study the following equation \begin{equation*} \frac{\partial
u(t,\,x)}{\partial t}= \Delta u(t,\,x)+b(u(t,\,x))+\sigma \dot{W}(t,\,x),\quad
t>0, \end{equation*} where $\sigma$ is a positive constant and $\dot{W}$ is a
space-time white noise. The initial condition $u(0,x)=u_0(x)$ is assumed to be
a nonnegative and continuous function. We first study the problem on $[0,\,1]$
with homogeneous Dirichlet boundary conditions. Under some suitable conditions,
together with a theorem of Bonder and Groisman, our first result shows that the
solution blows up in finite time if and only if \begin{equation*}
\int_{\cdot}^\infty\frac{1}{b(s)}\,d s<\infty, \end{equation*} which is the
well-known Osgood condition. We also consider the same equation on the whole
line and show that the above condition is sufficient for the nonexistence of
global solutions. Various other extensions are provided; we look at equations
with fractional Laplacian and spatial colored noise in $\mathbb{R}^d$.
