  We consider the heat equation with a superlinear absorption term
$\partial_{t} u-\Delta u= -u^{p}$ in $\mathbb{R}^n$ and study the existence and
nonexistence of nonnegative solutions with an $m$-dimensional time-dependent
singular set, where $n-m\geq 3$. First, we prove that if $p\geq (n-m)/(n-m-2)$,
then there is no singular solution. We next prove that, if $1<p<(n-m)/(n-m-2)$,
then there are two types of singular solution. Moreover, we show the uniqueness
of the solutions and specify the exact behavior of the solutions near the
singular set.
