  We study a branching random walk on $\r$ with an absorbing barrier. The
position of the barrier depends on the generation. In each generation, only the
individuals born below the barrier survive and reproduce. Given a reproduction
law, Biggins et al. \cite{BLSW91} determined whether a linear barrier allows
the process to survive. In this paper, we refine their result: in the boundary
case in which the speed of the barrier matches the speed of the minimal
position of a particle in a given generation, we add a second order term $a
n^{1/3}$ to the position of the barrier for the $n^\mathrm{th}$ generation and
find an explicit critical value $a_c$ such that the process dies when $aa_c$.
We also obtain the rate of extinction when $a < a_c$ and a lower bound on the
surviving population when $a > a_c$.
