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.



Identifying the nonlinear parabolic PDE $u_t = (u^2)_{xx}$.
Since $(u^2)\_{xx} = (2uu\_x)\_x$ this is (up to a rescaling) the [porous medium equation](http://mathworld.wolfram.com/PorousMediumEquation.html) for $m=1$.


