Searching for partially reactive sites

How do single or multiple (sub)diffusing particles search for a 
target with a partially reactive boundary?  A finite reaction rate 
which is typical for many chemical or biochemical reactions, is 
introduced as the possibility for a particle to find a target but 
not to "recognize" it.  Searching is not finished until the target 
is found and recognized.  For a single searching particle, the short 
and long-time regimes are investigated, with a special focus on 
comparison between perfectly and partially reactive boundaries.  
For multiple searching particles, explicit formulas for the probability 
density of the search time are given for subdiffusion in one and three 
dimensions.  The dependence of the mean search time on the density of 
particles and the reaction rate is analyzed.  Unexpectedly, in the 
high density limit, the particles undergoing slower subdiffusive 
motion find a target faster.