  We study a free boundary problem for a parabolic partial differential
equation in which the solution is coupled to the moving boundary through an
integral constraint. The problem arises as the hydrodynamic limit of an
interacting particle system involving branching Brownian motion with selection,
the so-called Brownian bees model which is studied in a companion paper. In
this paper we prove existence and uniqueness of the solution to the free
boundary problem, and we characterise the behaviour of the solution in the
large time limit.
