The logarithmic fast diffusion equation in one space variable with periodic boundary conditions. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:47:08Z http://mathoverflow.net/feeds/question/117664 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117664/the-logarithmic-fast-diffusion-equation-in-one-space-variable-with-periodic-bound The logarithmic fast diffusion equation in one space variable with periodic boundary conditions. Jeff Dodd 2012-12-30T20:20:31Z 2012-12-30T21:05:49Z <p>I need to know about this non-linear logarithmic fast diffusion equation for a function <code>$u(x,t)$</code> of one space variable <code>$x$</code> and time <code>$t$</code>: <code>$$u_t = (\ln u)_{xx}$$</code> which is to run on an interval <code>$a \leq x \leq b$</code> with periodic boundary conditions <code>$$u(a,t) = u(b,t)$$</code> <code>$$u_x(a,t) = u_x(b,t)$$</code> for <code>$t \geq 0$</code> and an initial condition <code>$$u(x,0) = f(x)$$</code><br> where <code>$f$</code> is a smooth, strictly positive function defined for <code>$a \leq x \leq b$</code> (which itself satisfies the above boundary conditions). </p> <p>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 <code>$t \rightarrow \infty$</code>. An extensive literature search has turned up similar results for similar problems, but nothing I can quote for this particular problem. </p> <p>QUESTION: Does anyone know of any literature that addresses existence, uniqueness, regularity, and\or behavior as <code>$t \rightarrow \infty$</code> of solutions of this initial-boundary-value problem? (Failing that, any advice or insight about this problem would be greatly appreciated!) </p>