  We show that the tail distribution $U$ of the explosion time for a
multidimensional diffusion (and more generally, a suitable function
$\mathscr{U}$ of the Feynman-Kac type involving the explosion time) is a
viscosity solution of an associated parabolic partial differential equation
(PDE), provided that the dispersion and drift coefficients of the diffusion are
continuous. This generalizes a result of Karatzas and Ruf (2013), who
characterize $U$ as a classical solution of a Cauchy problem for the PDE in the
one-dimensional case, under the stronger condition of local H\"older continuity
on the coefficients. Furthermore, we show that $\mathscr{U}$ is dominated by
any nonnegative classical supersolution of this Cauchy problem, and that
$\mathscr{U}$ is the smallest lower-semicontinuous viscosity supersolution of
that PDE with an appropriate boundary condition, provided it is a classical
solution. We also consider another notion of weak solvability, that of the
distributional (sub/super)solution, and show that $\mathscr{U}$ is no greater
than any nonnegative distributional supersolution of the relevant PDE. Finally,
we establish the joint continuity of $\mathscr{U}$ in the one-dimensional case.
