  We consider the Prandtl boundary layer equations on the half plane, with
initial datum that lies in a weighted $H^1$ space with respect to the normal
variable, and is real-analytic with respect to the tangential variable. The
boundary trace of the horizontal Euler flow is taken to be a constant. We prove
that if the Prandtl datum lies within $\varepsilon$ of a stable profile, then
the unique solution of the Cauchy problem can be extended at least up to time
$T_\varepsilon \geq \exp(\varepsilon^{-1}/ \log(\varepsilon^{-1}))$.
