Consider the PDE for $p:[0,T]\times [0,1]\to\mathbb R$ as follows:
$$
\begin{cases}
p_t = e^{-p}p_{xx}, & (t,x)\in (0,T)\times (0,1),\\
p(0,\cdot)\equiv -c &\text{for }x\in (0,1)\\
p(\cdot,0)\equiv 0 \equiv p(\cdot,1)&\text{for  }t\in (0,T),
\end{cases}
$$ 
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.