Cisco Webex, Online seminar
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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%2C%20%5Cqquad%20x%20%5Cin%20%7B%5Cbf%20R%7D%5EN%20%5Csetminus%20%5C%7B%20%5Cxi(t)%20%5C%7D&chf=bg,s,333333&chco=ffffff)
,
where
is typically given by
In the subcritical case where
and
%5E2%2F4%24&chf=bg,s,333333&chco=ffffff)
, assuming that the motion of the singularity $\xi(t)$ is not so quick (at least,
-Hölder continuous with
), we can show that there exist two types of positive solutions, and that the conditions on
and
are optimal. On the other hand, when the singularity moves like a fractional Brownian motion with the Hurst exponent
, there exists a positive solution for a wider range
.
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