Hot spots of solutions to the heat equation with inverse square potential

ABSTRACT

We investigate the large time behavior of the hot spots of the solution to the Cauchy problem $$\left\{\begin{array}{cc}{\partial }_{t}u-\mathrm{\Delta }u+V(|x|)u=0\hfill & \text{in}{\mathbf{R}}^N\times (0,\infty ),\hfill \\ u(x,0)=\phi (x)\hfill & \text{in}{\mathbf{R}}^N,\hfill \end{array}\right.$$

where $\phi \in L^2({\mathbf{R}}^N,e^{{|x|}^2/4}\mathrm{d}x)$ and $V=V(r)$ decays quadratically as $r\to \infty$. In this paper, based on the arguments in [K. Ishige and A. Mukai, to appear in Discrete Contin. Dyn. Syst.], we classify the large time behavior of the hot spots of u and reveal the relationship between the behavior of the hot spots and the harmonic functions for $-\mathrm{\Delta }+V$.

KEYWORDS:

• Hot spots
• heat equation
• inverse square potential

AMS Subject Classifications:

• 35K05
• 35K15

Acknowledgements

The authors of this paper would like to thank the referees for carefully reading the manuscript and relevant remarks.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was partially supported by the Grant-in-Aid for Scientific Research (A) [grant number 15H02058] from Japan Society for the Promotion of Science. The second author was supported in part by JSPS KAKENHI [grant number 15K04965], [grant number 15H03631).