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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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up vote 4 down vote accepted

Since $(u^2)_{xx} = (2uu_x)_x$ this is (up to a rescaling) the porous medium equation for $m=1$.

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See also en.wikipedia.org/wiki/… for questions of the form "is this a well-known PDE"? –  Steve Huntsman May 20 '10 at 22:48
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And for the PME, see books.google.com/books?id=4E2Gc67b-dYC –  Steve Huntsman May 20 '10 at 22:52
    
Thank you! This is a very useful source. –  John Jiang May 21 '10 at 0:45

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