  The problem of the time required for a diffusing molecule, within a large
bounded domain, to first locate a small target is prevalent in biological
modeling. Here we study this problem for a small spherical target. We develop
uniform in time asymptotic expansions in the target radius of the solution to
the corresponding diffusion equation. Our approach is based on combining
short-time expansions using pseudo-potential approximations with long-time
expansions based on first eigenvalue and eigenfunction approximations. These
expansions allow the calculation of corresponding expansions of the first
passage time density for the diffusing molecule to find the target. We
demonstrate the accuracy of our method in approximating the first passage time
density and related statistics for the spherically symmetric problem where the
domain is a large concentric sphere about a small target centered at the
origin.
