Consider the one-dimensional logarithmic diffusion equation for $u: \mathbb R_+ \times [0,1]\ni (t,x)\to u(t,x)\in \mathbb R$ : $$(\ast)\quad\quad \begin{cases} 2u_t = \big(\log(u)\big)_{xx} & \text{for }(t,x)\in (0,\infty)\times (0,1),\\ \\ u(0,x)= g(x) &\text{for }x\in (0,1),\\ \\ u(t,0)= 1= u(t,1)&\text{for }t\in (0,\infty), \end{cases} $$

where $g:[0,1]\to (0,\infty)$ is smooth s.t. $g(0)=1=g(1)$ and $g^{(k)}(0)=0=g^{(k)}(1)$ for all $k\ge 1$. 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 $(t,x)\in (0,\infty)\times [0,1]$. Does the classical solution exist? If so, is there a study on its regularity?

Any answer, comment and reference are highly appreciated.