  Context. DG Tau is a low-mass pre-main sequence star, whose strongly
accreting protoplanetary disk exhibits a so-far enigmatic behavior: its
mid-infrared thermal emission is strongly time-variable, even turning the 10
$\mu$m silicate feature from emission to absorption temporarily. Aims. We look
for the reason for the spectral variability at high spatial resolution and at
multiple epochs. Methods. We study the temporal variability of the mid-infrared
interferometric signal, observed with the VLTI/MIDI instrument at six epochs
between 2011 and 2014. We fit a geometric disk model to the observed
interferometric signal to obtain spatial information about the disk. We also
model the mid-infrared spectra by template fitting to characterize the profile
and time dependence of the silicate emission. We use physically motivated
radiative transfer modeling to interpret the mid-infrared interferometric
spectra. Results. The inner disk (r<1-3 au) spectra exhibit a 10 $\mu$m
absorption feature related to amorphous silicate grains. The outer disk (r>1-3
au) spectra show a crystalline silicate feature in emission, similar to the
spectra of comet Hale-Bopp. The striking difference between the inner and outer
disk spectral feature is highly unusual among T Tauri stars. The mid-infrared
variability is dominated by the outer disk. The strength of the silicate
feature changed by more than a factor of two. Between 2011 and 2014 the
half-light radius of the mid-infrared-emitting region decreased from 1.15 to
0.7 au. Conclusions. For the origin of the absorption we discuss four possible
explanations: a cold obscuring envelope, an accretion heated inner disk, a
temperature inversion on the disk surface and a misaligned inner geometry. The
silicate emission in the outer disk can be explained by dusty material high
above the disk plane, whose mass can change with time, possibly due to
turbulence in the disk.
