  We discuss mesons in thermalizing gluon backgrounds in the N=2 supersymmetric
QCD using the gravity dual. We numerically compute the dynamics of a probe
D7-brane in the Vaidya-AdS geometry that corresponds to a D3-brane background
thermalizing from zero to finite temperatures by energy injection. In static
backgrounds, it has been known that there are two kinds of brane embeddings
where the brane intersects the black hole or not. They correspond to the phases
with melted or stable mesons. In our dynamical setup, we obtain three cases
depending on final temperatures and injection time scales. The brane stays
outside of the black hole horizon when the final temperature is low, while it
intersects the horizon and settles down to the static equilibrium state when
the final temperature is high. Between these two cases, we find the overeager
case where the brane dynamically intersects the horizon although the final
temperature is not high enough for a static brane to intersect the horizon. The
interpretation of this phenomenon in the dual field theory is meson melting due
to non-thermal effects caused by rapid energy injection. In addition, we
comment on the late time evolution of the brane and a possibility of its
reconnection.
