Consider the PDE for $p:[0,T]\times [0,1]\to\mathbb R$ as follows:

$$p_t = e^{-p}p_{xx},\quad (t,x)\in (0,T)\times (0,1),$$

$p(0,\cdot)\equiv -c$ for $x\in (0,1)$ and $p(\cdot,0)\equiv 0 \equiv p(\cdot,1)$ for  $t\in (0,T)$, where $c>0$ is some constant. Is there any wellposedness result on this initial-boundary problem? In particular, can we have the regularity of $p$ on $(0,T)\times (0,1)$? 

Any answers, comments and references are highly appreciated.

PS: I did a search in Partial Differential Equations (Taylor), while nothing interesting has been found up to now.