  We consider, through PDE methods, branching Brownian motion with drift and
absorption. It is well know that there exists a critical drift which separates
those processes which die out almost surely and those which survive with
positive probability. In this work, we consider lower order terms to the
critical drift which ensures a non-negative, bounded expected number of
particles and convergence of this expectation to a limiting number,
$\alpha_0\geq 0$, which is positive for some initial data. In particular, we
show that, in an average sense, the expected number of particles stabilizes to
$\alpha_0$ like $O(\log(t)/t)$ if and only if the multiplicative factor of the
$O(t^{-1/2})$ correction term is $3\sqrt{\pi} t^{-1/2}$. Otherwise, the
convergence is like $O(1/\sqrt{t})$. We point out some connections between this
work and recent work investigating the expansion of the front location for the
initial value problem in Fisher-KPP.
