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

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:

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.