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 $m=1$. 

