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.

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.