Motivated by the study of branching particle systems with selection, we establish global existence for the solution (u, µ) of the free boundary problem          ∂ t u = ∂ 2 x u + u for t > 0 and x > µ t , u(x, t) = 1 for t > 0 and x ≤ µ t , ∂ x u(µ t , t) = 0 for t > 0, u(x, 0) = v(x) for x ∈ R, when the initial condition v : R → [0, 1] is non-increasing with v(x) → 0 as x → ∞ and v(x) → 1 as x → −∞. We construct the solution as the limit of a sequence (u n) n≥1 , where each u n is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term. Recent results of De Masi et al. [5] show that this global solution can be identified with the hydrodynamic limit of the so-called N-BBM, i.e. a branching Brownian motion in which the population size is kept constant equal to N by killing the leftmost particle at each branching event.