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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 $$u(au(a,t) = u(bu(b,t)$$ $$u_x(au_x(a,t) = u_x(bu_x(b,t)$$ 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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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 $$u(a) = u(b)$$ $$u_x(a) = u_x(b)$$ 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!)