  The aim of this paper is to analyze an SPDE which arises naturally in the
context of Liouville quantum gravity. This SPDE shares some common features
with the so-called Sine-Gordon equation and is built to preserve the Liouville
measure which has been constructed recently on the two-dimensional sphere $S^2$
and the torus $T^2$ in the work by David-Kupiainen-Rhodes-Vargas. The SPDE we
shall focus on has the following (simplified) form: \[ p_t X = \frac 1 {4\pi}
\Delta X - e^{\gamma X} + \xi\,, \] where $\xi$ is a space-time white noise on
$R_+ \times S^2$ or $R_+ \times T^2$. The main aspect which distinguishes this
singular stochastic SPDE with well-known SPDEs studied recently (KPZ, dynamical
$\Phi^4_3$, dynamical Sine-Gordon, etc.) is the presence of intermittency. One
way of picturing this effect is that a naive rescaling argument suggests the
above SPDE is sub-critical for all $\gamma>0$, while we don't expect solutions
to exist when $\gamma > 2$. In this work, we initiate the study of this
intermittent SPDE by analyzing what one might call the "classical" or "Da
Prato-Debussche" phase which corresponds here to
$\gamma\in[0,\gamma_{dPD}=2\sqrt{2} -\sqrt{6})$. By exploiting the positivity
of the non-linearity $e^{\gamma X}$, we can push this classical threshold
further and obtain this way a weaker notion of solution when
$\gamma\in[\gamma_{dPD}, \gamma_{pos}=2\sqrt{2} - 2)$. Our proof requires an
analysis of the Besov regularity of natural space/time Gaussian multiplicative
chaos (GMC) measures. Regularity Structures of arbitrary high degree should
potentially give strong solutions all the way to the same threshold
$\gamma_{pos}$ and should not push this threshold further. Of independent
interest, we prove along the way (using techniques from [HS16]) a stronger
convergence result for approximate GMC measures which holds in Besov spaces.
