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
$$ u(a,t) = u(b,t) $$
$$ u\_x(a,t) = u\_x(b,t) $$
 for $t \geq 0$ 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!) 



The logarithmic fast diffusion equation in one space variable with periodic boundary conditions.
Maybe this book is a good choice: <http://ukcatalogue.oup.com/product/9780198569039.do#.UaC1zPER0uU>


In the web of the author (<http://www.uam.es/personal_pdi/ciencias/jvazquez/coursejlv.html>) you can download the index, preface and introductory chapter.


I hope this helps you even if it's a late response.


