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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:
![](https://chart.googleapis.com/chart?cht=tx&chl=u_t%3D%5CDelta%20u%20%2B%20V(x%2Ct)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
![](https://chart.googleapis.com/chart?cht=tx&chl=%20%24V%24&chf=bg,s,333333&chco=ffffff)
is typically given by
In the subcritical case where
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmu%3D2%24&chf=bg,s,333333&chco=ffffff)
and
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Clambda%20%3C%20(N-2)%5E2%2F4%24&chf=bg,s,333333&chco=ffffff)
, assuming that the motion of the singularity $\xi(t)$ is not so quick (at least,
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cgamma%24&chf=bg,s,333333&chco=ffffff)
-Hölder continuous with
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cgamma%3E1%2F2%24&chf=bg,s,333333&chco=ffffff)
), we can show that there exist two types of positive solutions, and that the conditions on
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Clambda%24&chf=bg,s,333333&chco=ffffff)
and
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmu%24&chf=bg,s,333333&chco=ffffff)
are optimal. On the other hand, when the singularity moves like a fractional Brownian motion with the Hurst exponent
![](https://chart.googleapis.com/chart?cht=tx&chl=%24H%3C1%2F2%24&chf=bg,s,333333&chco=ffffff)
, there exists a positive solution for a wider range
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmu%20%3C%20%5Cmin%20%5C%7B%20H%2C%201%2FN%20%5C%7D%24&chf=bg,s,333333&chco=ffffff)
.
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