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 __\( \mathbb{R}^3 \)__.