Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 13.10.2016 at 14:00 Lecture room M-223

Jin Takahashi (Tokyo Institute of Technology):

Time-dependent singularities in a semilinear heat equation

Abstract:

We consider the following semilinear heat equation

$u_t-\Delta u=u^p$, $x\in \mathbf{R}^N\setminus\{\xi(t)\}$, $t\in I$,

where $N\geq3$, $p>1$, $I\subset\mathbf{R}$ is an open interval and

$\xi:\overline{I}\rightarrow\mathbf{R}^N$ is a prescribed curve which is smooth enough.

The aim of this talk is to study time-dependent singularities of nonnegative solutions

of the above equation under the assumption that $1<p<N/(N-2)$.

In the first part of this talk, we prove that every solution $u$

can be extended as a distributional solution of the following equation

$u_t-\Delta u=u^p +(\delta_0\otimes\mu)\circ\mathcal{T}$ in ${\cal D}'(\mathbf{R}^N \times I)$.

Here $\mathcal{T} (\varphi) (x,t):=\varphi(x+\xi(t),t)$, $\delta_0$ is the Dirac measure on $\mathbf{R}^N$

concentrated at the origin and $\mu$ is a Radon measure on $I$ determined by the solution $u$.

In addition, we show relations between the exponent $p$ and the local growth rate of $\mu$

and specify the behavior of solutions at the time-dependent singularity.

In the second part of this talk, we give sharp conditions on $\mu$ for the existence and the nonexistence

of solutions of the above extended equation.

This is a joint work with Dr. Toru Kan (Tokyo Institute of Technology).