Let \( D \) be a bounded open set containing the origin, having \( C^2 \) boundary and with diameter less than or equal to one. For each \( N = 1,2,\dots \), let \( y_1^{(N)}, y_2^{(N)},\dots,y_N^{(N)} \) be points in \( \mathbb{R}^3 \) and define sets \( D_i^{(N)} \) by
\[ D_i^{(N)} = \bigl\{x\in \mathbb{R}^3\mid N(x-y_i^{(N)})\in D\bigr\}, \]
\( i = 1,2,\ldots,N \). We shall call the set \( D_i^{(N)} \) the hole centered at \( y_i^{(N)} \) with diameter less than or equal to \( N^{-1} \). Let \( G^{(N)} \) denote the region
\[ G^{(N)} = \mathbb{R}^3-\bigcup_{i=1}^N D_i^{(N)} \]
which is \( \mathbb{R}^3 \) with holes of diameter \( \leq N^{-1} \) centered at \( y_1^{(N)},\dots,y_N^{(N)} \). We shall analyze the asymptotic behavior of \( u^{(N)}(x,t) \) as \( N\to\infty \) which is the solution of
\begin{align*}
\frac{\partial}{\partial t} u^{(N)}(x,t) &= \frac{1}{2}\Delta u^{(N)}(x,t), && t > 0,\ x\in G^{(N)},\\
u^{(N)}(x,t) &=0, && t > 0,\ x\in\partial G^{(N)} = \bigcup_{i=1}^N\partial D_i^{(N)},\\
u^{(N)}(x,0) &=f(x), && x\in G^{(N)}, \end{align*}
with \( f(x) \) a given bounded continuous function with compact support in \( \mathbb{R}^3 \).
