Solutions with time-dependent singular sets for the heat equation with absorption

We consider the heat equation with a superlinear absorption term ${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=-u^p$ in ${\mathbb{R}}^n$ and study the existence and nonexistence of nonnegative solutions with an $m$-dimensional time-dependent singular set, where $n-m\ge 3$. First, we prove that if $p\ge (n-m)\mathrm{/}(n-m-2)$, then there is no singular solution. We next prove that, if $1<p<(n-m)\mathrm{/}(n-m-2)$, 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.