# The logarithmic fast diffusion equation in one space variable with periodic boundary conditions.

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!)

• Have you tried to mimic other proofs for similar equations? Dec 30, 2012 at 21:08
• Not yet, Deane. I suspect that other proofs for similar equations could be adapted, and I may have to go that route. But for my purposes, it would be much better if I did not have to make such a technical digression. It is a natural enough problem that I am hoping someone might have treated it somewhere so that I can just say what I need to say. -- Jeff Dodd Dec 30, 2012 at 22:37
• Diffusion is fast only for $u\to 0$. In your problem, however, the maximum principle ensures that the solution remains within the range of the initial data. This makes the analysis fairly routine. Dec 30, 2012 at 23:59
• Changing $u=e^v$ gets $v_t=e^{-v}v_{xx}$. I haven't a reference but general results for weakly nonlinear equations of the form $v_t=a(v)v_{xx}$ should apply. Jan 5, 2013 at 11:34