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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.

flag	asked May 20 2010 at 22:24 John Jiang 1,846●3●11

Steve Huntsman · Accepted Answer · 2010-05-20 22:46:59Z UTC

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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

Steve Huntsman
8,139●1●22●56

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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 2010 at 22:48

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And for the PME, see books.google.com/books?id=4E2Gc67b-dYC – Steve Huntsman May 20 2010 at 22:52

Thank you! This is a very useful source. – John Jiang May 21 2010 at 0:45

Identifying the nonlinear parabolic PDE $u_t = (u^2)_{xx}$.

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