Consider the one-dimensional *logarithmic diffusion equation* for $u:  \mathbb R_+ \times [0,1]\ni (t,x)\to u(t,x)\in \mathbb R$ : 
$$(\ast)\quad\quad 
\begin{cases}
2u_t = \big(\log(u)\big)_{xx} & \text{for }(t,x)\in (0,\infty)\times (0,1),\\
\\
u(0,x)= 0 &\text{for }x\in (0,1),\\
\\
u(t,0)= 1= u(t,1)&\text{for  }t\in (0,\infty).
\end{cases}
$$ 
In view of [*The nonlinear diffusion limit for generalized Carleman models: the initial-boundary value problem*][1] the problem $(\ast)$ admits a unique "weak solution" denoted by $u$ (see the above reference for its definition) such that $0<u(t,x)\le 1$ 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, comment and reference are highly appreciated.



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