Asymptotics for the heat equation in the exterior of a shrinking compact set in the plane via Brownian hitting times
HTML articles powered by AMS MathViewer
- by Ross G. Pinsky PDF
- Proc. Amer. Math. Soc. 130 (2002), 1673-1679 Request permission
Abstract:
Let $D_{r}=\{x\in R^{2}:|x|\le r\}$ 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{equation*} \begin {split} &u_{t}=\frac {1}{2}\Delta u, \quad x\in R^{2}-D_{\gamma (t)}, \quad t>0,\\ &u(x,0)=0,\quad x\in R^{2}-D_{\gamma (t)},\\ &u(x,t)=1,\quad x\in D_{\gamma (t)}, t>0. \end{split} \end{equation*} 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 $L_{r}=\{(x_{1},0)\in R^{2}:|x_{1}|\le r\}$. Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.References
- Ross G. Pinsky, Positive harmonic functions and diffusion, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995. MR 1326606, DOI 10.1017/CBO9780511526244
- Rogers, L.C.G. and Williams, D., Diffusions, Markov Processes and Martingales, Vol. 1, 2nd ed., Cambridge Univ. Press, 2000.
- Frank Spitzer, Some theorems concerning $2$-dimensional Brownian motion, Trans. Amer. Math. Soc. 87 (1958), 187–197. MR 104296, DOI 10.1090/S0002-9947-1958-0104296-5
Additional Information
- Ross G. Pinsky
- Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000 Israel
- Email: pinsky@techunix.technion.ac.il
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1673-1679
- MSC (1991): Primary 35K05, 35B40, 60J65
- DOI: https://doi.org/10.1090/S0002-9939-01-06206-2
- MathSciNet review: 1887014