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	NCTS Webinar on Nonlinear Evolutionary Dynamics

13:30 - 15:00, November 16, 2021 (Tuesday)
Cisco Webex, Online seminar
(線上演講 Cisco Webex)
On the Heat Equation with a Dynamic Singular Potential
Eiji Yanagida (Tokyo Institute of Technology)

Abstract:

Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a Hardy-type singular potential:

$u_t=\Delta u + V(x,t)u, \qquad x \in {\bf R}^N \setminus \{ \xi(t) \}$ ,

where $V$ is typically given by

$V(x,t) = \frac{\lambda}{|x - \xi(t)|^\mu} \qquad (\lambda>0, \ \mu>0)$

In the subcritical case where $\mu=2$ and $\lambda < (N-2)^2/4$ , assuming that the motion of the singularity $\xi(t)$ is not so quick (at least, $\gamma$ -Hölder continuous with $\gamma>1/2$ ), we can show that there exist two types of positive solutions, and that the conditions on $\lambda$ and $\mu$ are optimal. On the other hand, when the singularity moves like a fractional Brownian motion with the Hurst exponent $H<1/2$ , there exists a positive solution for a wider range $\mu < \min \{ H, 1/N \}$ .

JOIN WEBEX MEETING

https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m1653c816d243503555c58d00842549d4

Meeting number (access code): 2516 907 0813

Meeting password: hqQGMViJ276 (47746845 from phones and video systems)