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

Jin Takahashi; Hikaru Yamamoto

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arXiv · 1712.06065

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

Jin Takahashi·Hikaru Yamamoto

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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 $n-m\geq 3$. 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.

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Jin Takahashi, Hikaru Yamamoto. 2017-12-17. Solutions with time-dependent singular sets for the heat equation with absorption. https://arxiv.org/abs/1712.06065

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