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.