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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!)

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  • $\begingroup$ Have you tried to mimic other proofs for similar equations? $\endgroup$
    – Deane Yang
    Commented Dec 30, 2012 at 21:08
  • $\begingroup$ 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 $\endgroup$
    – Jeff Dodd
    Commented Dec 30, 2012 at 22:37
  • $\begingroup$ 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. $\endgroup$
    – Michael Renardy
    Commented Dec 30, 2012 at 23:59
  • $\begingroup$ 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. $\endgroup$
    – Andrew
    Commented Jan 5, 2013 at 11:34

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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.

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I should have updated my post here some time ago. I can attest that Rafa's answer to my question identifies exactly the right trail to follow. It turns out that J.L. Vazquez has done comprehensive work on the logarithmic diffusion equation that (once I found it!) provided me with just about all I needed to know about it.

The particular paper that allowed me to address the above issue and get on with things is "A Nonlinear Heat Equation with Singular Diffusivity" by J.R. Esteban, A. Rodriguez, and J.L. Vazquez, Communications in Partial Differential Equations, 13(8), 985-1039 (1988). (Periodic initial data is not explicitly mentioned there, or anywhere else that I have found, but the more general case of smooth, positive initial data that is bounded and bounded away from 0 is addressed in detail, and armed with those results I was able to confirm that everything I wanted to be able to say in the special case of periodic initial data is true.)

A brief general discussion of logarithmic diffusion in one dimension with many useful references is given in Chapter 8 of the book Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type by J.L. Vazquez, Oxford University Press, 2006, which is a companion book to the one Rafa has cited in his answer.

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