  We consider the kinetics of the imperfect narrow escape problem, i.e. the
time it takes for a particle diffusing in a confined medium of generic shape to
reach and to be adsorbed by a small, imperfectly reactive patch embedded in the
boundary of the domain, in two or three dimensions. Imperfect reactivity is
modeled by an intrinsic surface reactivity $\kappa$ of the patch, giving rise
to Robin boundary conditions. We present a formalism to calculate the exact
asymptotics of the mean reaction time in the limit of large volume of the
confining domain. We obtain exact explicit results in the two limits of large
and small reactivities of the reactive patch, and a semi-analytical expression
in the general case. Our approach reveals an anomalous scaling of the mean
reaction time as the inverse square root of the reactivity in the large
reactivity limit, valid for an initial position near the extremity of the
reactive patch. We compare our exact results with those obtained within the
``constant flux approximation''; we show that this approximation turns out to
give exactly the next-to-leading order term of the small reactivity limit, and
provides a good approximation of the reaction time far from the reactive patch
for all reactivities, but not in the vicinity of the boundary of the reactive
patch due to the above mentioned anomalous scaling. These results thus provide
a general framework to quantify the mean reaction times for the imperfect
narrow escape problem.
