## Seminar 5.3.2020: Jin Takahashi

Seminar on Qualitative Theory of Differential Equations
organized by P.Quittner, M.Fila and R.Kollar

Moderator: sevcovic

### Seminar 5.3.2020: Jin Takahashi

Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations

Thursday 5.3.2020 at 14:00 Lecture room M-223

Jin Takahashi (Tokyo Institute of Technology):
Existence of solutions with moving singularities
for a semilinear heat equation with a critical exponent

Abstract:
We consider nonnegative singular solutions of a superlinear heat equation $u_t -\Delta u=u^p$ on $\mathbf{R}^N$.
Sato-Yanagida (2009) showed the existence of solutions with moving singularities in the case $N/(N-2)<p<p_{JL}^*$,
where $p_{JL}^*$ is an explicit exponent. Kan-T. (2017) studied the case $p<N/(N-2)$.
In this talk, we construct a solution with an explicit moving singularity in a critical case $p=N/(N-2)$.
In the context of the uniqueness of mild solutions, our result refines counter examples given by Terraneo (2002)
in view of pointwise behavior.
quittner

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