Consider the non-linear PDE for $u:[0,1]\times [-1,1]\to\mathbb R$ as follows:
$$u_t= \inf_{b\ge 1/e} \big(-b u_{xx} - \log b - 1\big), \quad \forall (t,x) \in (0,1) \times (-1,1),$$
together with the terminal and boundary conditions $u(1,\cdot) = 0=u(\cdot,\pm 1)$. Is there any result on the existence/uniqueness of the (classical or generalized) solution? If so, what can we say about the regularity of the solution on $(0,1) \times (-1,1)$?
Any answer, comments or references are highly appreciated.
