  The thermodynamics and the kinetics of Hawking-Page phase transition were
studied previously based on the free energy landscape. The AdS black hole can
evaporate if imposing the absorbing boundary conditions at infinity. We suggest
that the kinetics of Hawking-Page phase transition should be governed by the
reaction-diffusion equation, where the Hawking evaporation plays the role of
the reaction on the background of the free energy landscape. By calculating the
mean first passage time from the large black hole phase to the thermal gas
phase, we show that the phase transition can occur more easily under the
Hawking radiation. In particular, a kinetic turnover is observed when
increasing the ensemble temperature or the frictions. This kinetic turnover can
be viewed as the dynamical phase transition to identify the time scale where
Hawking evaporation process is comparable to Hawking-Page phase transition.
