We study a white-noise driven semilinear partial
differential equation on the spatial interval $[0,1]$ with
Dirichlet boundary condition and with a singular drift
of the form $c u^{-3}$, $c>0$. We prove
existence and uniqueness of a non-negative continuous adapted
solution $u$ on $[0,\infty)\times[0,1]$ for every
nonnegative continuous initial datum $x$, satisfying
$x(0)=x(1)=0$.
We prove that the law $\pi_\delta$ of the Bessel bridge
on $[0,1]$ of dimension $\delta>3$ is the unique invariant
probability measure of the process $x\mapsto u$, with
$c=(\delta-1)(\delta-3)/8$ and, if $\delta\in{\mathbb N}$,
that $u$ is the radial part in the sense of Dirichlet forms
of the ${\mathbb R}^\delta$-valued solution of a linear stochastic heat equation.
An explicit integration by parts formula w.r.t. $\pi_\delta$ is given for all $\delta>3$.