Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 12.10.2017 at 15:00 Lecture room M-223

Jin Takahashi (Tohoku University):

Solutions with a higher dimensional singular set for the heat equation with absorption

Abstract:

In this talk, we consider nonnegative solutions with a time-dependent $m$-dimensional singular set

for the linear heat equation in $\mathbf{R}^n$ with an absorption term $-u^p$.

Here the conditions $n-m\geq3$ and $p>1$ and some assumption on the singular set are imposed.

We show that if $p\geq (n-m)/(n-m-2)$, there is no singular solution.

On the other hand, in the case $p< (n-m)/(n-m-2)$, we construct two types of global-in-time singular solution.

We also show the uniqueness of solutions in some class.

This is a joint work with Hikaru Yamamoto (Tokyo University of Science).