I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$: $$ u_t = (\ln u)_{xx}$$ which is to run on an interval $ a \leq x \leq b $ with periodic boundary conditions $$ u(a) = u(b) $$ $$ u_x(a) = u_x(b) $$ and an initial condition $$ u(x,0) = f(x) $$
where $f$ is a smooth, strictly positive function defined for $ a \leq x \leq b $ (which itself satisfies the above boundary conditions).

In particular, I would like to be able to say that there is a smooth solution of this initial-boundary-value problem which approaches the constant equilibrium solution as $t \rightarrow \infty$. An extensive literature search has turned up similar results for similar problems, but nothing I can quote for this particular problem.

QUESTION: Does anyone know of any literature that addresses existence, uniqueness, regularity, and\or behavior as $t \rightarrow \infty$ of solutions of this initial-boundary-value problem? (Failing that, any advice or insight about this problem would be greatly appreciated!)