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

Abstract

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 $n-m\ge 3$. First, we prove that if $p\ge (n-m)/(n-m-2)$, 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.

arXiv comment47 pages, 4 figures

References in corpus (1)

• A Compactness Theorem for the Second Fundamental Form