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 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, comments and references are highly appreciated.