Asymptotics for the heat equation in the exterior of a shrinking compact set in the plane via Brownian hitting times (Q2781337)

The author considers the following system of equations NEWLINE\[NEWLINE\begin{cases} u_t= \frac 12\Delta u,\;x\in \mathbb{R}^2-D_{\gamma (t)},\;t>0,\\ u(x,0)=0,\;x\in \mathbb{R}^2-D_{\gamma (t)},\\ u(x,t)=1,\;x\in D_{\gamma(t)},\;t\geq 0,\end{cases}\tag{*}NEWLINE\]NEWLINE where \(D_r=\{x\in\mathbb{R}^2 :|x|<r\}\), and \(\gamma\) is a continuous, nonincreasing function on \([0,+\infty)\) satisfying \(\lim_{t\to \infty}\gamma (t)=0\). The author proves the following main theorem:NEWLINENEWLINENEWLINELet \(u(x,t)\) be the solution to (*). Assume that there exist constants \(0<c_1<c_2\) and a constant \(k>0\) such that \(c_1t^{-k}\leq \gamma(t) \leq c_2t^{-k}\) for sufficiently large \(t\). Then \(\lim_{t\to \infty}u (x,t)={1\over 1+2k}\).