A friend of mine in the department needs to know if the following PDE has been extensively studied $$ u_t = (u^2)_{xx}$$ Or more generally, replacing the square by any function of $u$. One would like to know uniqueness and existence of its solution, and smoothness property, for example.
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Since $(u^2)_{xx} = (2uu_x)_x$ this is (up to a rescaling) the porous medium equation for $m=1$.
answered May 20, 2010 at 22:46
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4$\begingroup$ See also en.wikipedia.org/wiki/… for questions of the form "is this a well-known PDE"? $\endgroup$ Commented May 20, 2010 at 22:48
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1$\begingroup$ And for the PME, see books.google.com/books?id=4E2Gc67b-dYC $\endgroup$ Commented May 20, 2010 at 22:52