  We study an injection-branching-selection particle system on $\R$ at the
hydrodynamic limit under arbitrarily varying injection and removal rates, where
the corresponding free boundary problem (FBP) is not in general known to be
solvable in the classical sense. We propose a weak formulation that does not
involve the notion of a free boundary, but reduces to a FBP when classical
solutions exist. It is based on second order parabolic equations with
measure-valued right-hand side in conjunction with a complementarity condition.
We show that the weak formulation characterizes the limit. We also study a
branching-selection model of motionless particles with nonlocal branching in
$\R^d$ under a general fitness function. The corresponding integro-differential
FBP, shown to characterize the hydrodynamic limit, is an (irreducible)
multi-dimensional evolution equation. In both results the treatment is based on
FBP uniqueness.
