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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotics for the heat equation in the exterior of a shrinking compact set in the plane via Brownian hitting times


Author: Ross G. Pinsky
Journal: Proc. Amer. Math. Soc. 130 (2002), 1673-1679
MSC (1991): Primary 35K05, 35B40, 60J65
Published electronically: October 5, 2001
MathSciNet review: 1887014
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Abstract: Let $D_{r}=\{x\in R^{2}:\vert x\vert\le r\}$ and let $\gamma $ be a continuous, nonincreasing function on $[0,\infty )$ satisfying $\lim _{t\to \infty }\gamma (t)=0$. Consider the heat equation in the exterior of a time-dependent shrinking disk in the plane:

\begin{displaymath}\begin{split} &u_{t}=\frac{1}{2}\Delta u, x\in R^{2}-D_{\gamm... ...\gamma (t)},\\ &u(x,t)=1, x\in D_{\gamma (t)}, t>0.\end{split}\end{displaymath}

If there exist constants $0<c_{1}<c_{2}$ and a constant $k>0$ such that $c_{1}t^{-k}\le \gamma (t)\le c_{2}t^{-k}$, for sufficiently large $t$, then $\lim _{t\to \infty }u(x,t)=\frac{1}{1+2k}$. The same result is also shown to hold when $D_{\gamma (t)}$is replaced by $L_{\gamma (t)}$, where $L_{r}=\{(x_{1},0)\in R^{2}:\vert x_{1}\vert\le r\}$. Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.


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Additional Information

Ross G. Pinsky
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000 Israel
Email: pinsky@techunix.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06206-2
PII: S 0002-9939(01)06206-2
Keywords: Heat equation, planar Brownian motion, hitting times, modulus of Brownian motion, large time asymptotics
Received by editor(s): May 20, 2000
Received by editor(s) in revised form: November 22, 2000
Published electronically: October 5, 2001
Additional Notes: This research was supported by the Fund for the Promotion of Research at the Technion
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2001 American Mathematical Society