  We consider the following stochastic heat equation \begin{equation*}
  \partial_t u(t\,,x) = \tfrac12 \partial^2_x u(t\,,x) + b(u(t\,,x)) +
\sigma(u(t\,,x)) \dot{W}(t\,,x), \end{equation*} defined for
$(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, where $\dot{W}$ denotes space-time
white noise. The function $\sigma$ is assumed to be positive, bounded, globally
Lipschitz, and bounded uniformly away from the origin, and the function $b$ is
assumed to be positive, locally Lipschitz and nondecreasing. We prove that the
Osgood condition \[
  \int_1^\infty\frac{\mathrm{d} y}{b(y)}<\infty \] implies that the solution
almost surely blows up everywhere and instantaneously, In other words, the
Osgood condition ensures that $\mathbb{P}\{ u(t\,,x)=\infty\quad\text{for all
$t>0$ and $x\in\mathbb{R}$}\}=1.$ The main ingredients of the proof involve a
hitting-time bound for a class of differential inequalities (Remark 4.3), and
the study of the spatial growth of stochastic convolutions using techniques
from the Malliavin calculus and the Poincar\'e inequalities that were developed
in Chen et al [3,4].
