  Boundary value problems for diffusion in singularly perturbed domains
(domains with small holes removed from the interior) is a topic of considerable
current interest. Applications include intracellular diffusive transport and
the spread of pollutants or heat from localized sources. In a previous paper,
we introduced a new method for characterizing the approach to steady-state in
the case of two-dimensional (2D) diffusion. This was based on a local measure
of the relaxation rate known as the accumulation time $T(\x)$. The latter was
calculated by solving the diffusion equation in Laplace space using a
combination of matched asymptotics and Green's function methods. We thus
obtained an asymptotic expansion of $T(\x)$ in powers of $\nu=-1/\ln \epsilon$,
where $\epsilon$ specifies the relative size of the holes. In this paper, we
develop the corresponding theory for three-dimensional (3D) diffusion. The
analysis is a non-trivial extension of the 2D case due to differences in the
singular nature of the Laplace transformed Green's function. In particular, the
asymptotic expansion of the solution of the 3D diffusion equation in Laplace
space involves terms of order $O((\epsilon/s)^n)$, where $s$ is the Laplace
variable. These $s$-singularities have to be removed by partial series
resummations in order to obtain an asymptotic expansion of $T(\x)$ in powers of
$\epsilon$.
