Solutions with time-dependent singular sets for the heat equation with absorption (Q1710732)

The authors consider nonnegative solutions of the heat equation with absorption term \[ \partial_tu-\Delta u=-|u|^{p-1}u\quad \text{in}\;(\Omega\times I)\setminus \widetilde{M}, \] where \(p>1,\) \(\Omega\subset\mathbb{R}^n\) is a domain, \(I\) is an open interval, and \(\widetilde{M}=\bigcup_{t\in I}(M_t\times\{t\})\) with an one parameter family \(\{M_t\}_{t\in I}\) of compact submanifolds of \(\Omega\) of dimension \(m\geq 1\) and codimension \(n-m\geq 3.\) It is proved that if \(1 < p < \frac{n-m}{n-m-2},\) then there exist solutions satisfying \(\lim_{x\to M_t}u(x,t)=+\infty\) for each \(t\in I.\) Moreover, uniqueness of such singular solutions is proved and their behaviour near the singular set \(\widetilde{M}\) is described.