Consider the one dimensional logarithmic diffusion equation for $u:  \mathbb R_+ \times [0,1]\ni (t,x)\to u(t,x)\in \mathbb R$ by 
$$
\begin{cases}
2u_t = \big(\log(u)\big)_{xx} & \text{for }(t,x)\in (0,\infty)\times (0,1),\\
u(0,x)= 0 &\text{for }x\in (0,1),\\
u(t,0)= 1= u(t,1)&\text{for  }t\in (0,\infty).
\end{cases}
$$ 
As a consequence of [*The nonlinear diffusion limit for generalized Carleman models: the initial-boundary value problem*][1] this initial-boundary value problem admits a unique "weak solution" denoted by $u$ (see the above reference for its definition) s.t. $u(t,x)>0$ for almost every $(t,x)\in (0,\infty)\times [0,1]$. My questions are as follows:

 1. Does the classical solution exist? If so, is there a study on its regularity?
 2. For every $\epsilon\in (0,1)$, is there a constant $c\equiv c(\epsilon)>0$ such that $\inf_{(t,x)\in [\epsilon,1]\times [0,1]}u(t,x)\ge c$?

Any answer, comments and references are highly appreciated.



  [1]: https://iopscience.iop.org/article/10.1088/0951-7715/20/4/007