On a non-linear PDE $p_t = e^{-p}p_{xx}$

Question

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.

Answer 1

Maybe a possibility would be to first get rid of the non-linearity by defining $\phi[\tilde{p}]$ to be the solution of : $$ \begin{cases} p_t = e^{-\tilde{p}}p_{xx}, & (t,x)\in (0,T)\times (0,1),\\ p(0,\cdot)\equiv -c &\text{for }x\in (0,1)\\ p(\cdot,0)\equiv 0 \equiv p(\cdot,1) &\text{for }t\in (0,T), \end{cases} $$ and then try to find a $p$ that satisfies $\phi[p]=p$ by some fixed-point arguments.

For one, if $p$ was a continuous solution of the original problem, then by the comparaison principle, $-c\leq p\leq 0$. So the space to study could be the space of continuous functions such that $p(\cdot,0)=p(\cdot,1)=0$ and $-c\leq p\leq 0$.

In that case, $e^{-\tilde{p}}$ would be very well bounded and behaved, so all the usual existence and regularity results apply (section 7.1 of "Partial Differential Equations" by Evans can pretty much be applied directly). To me, the main difficulty from there is proving the continuity of $\phi$ or similar properties to apply fixed-point theorems. I don't know of many estimates that explicitly involve the diffusion coefficients.