# A simple looking nonlinear PDE

A nonlinear PDE that has been bugging me for months, hoping someone has an idea, it looks so simple! Consider the following, for $u(x,t)$

$D \displaystyle\frac{\partial u}{\partial t} = u^2 \displaystyle\frac{\partial^2 u}{\partial x^2}$

subject to boundary conditions

$u(1,t) = 1$,

$\displaystyle\frac{\partial u}{\partial x} (1,t)= t$,

and initial condition

$u(x,0) =1$.

The standard similarity solution $u(x,t) = \displaystyle t^{\alpha}F\left(\frac{x}{ t^{\beta}}\right)$ fails due to the second boundary condition giving the same equation for $\alpha$ and $\beta$ as the PDE.

Any thoughts on linearising this, or other methods to obtain a solution would be much appreciated!

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I have just found a similar question at math.stackexchange.com/questions/107488/…. My apologies. I will leave this question as it stands just in case anyone has any new ideas to add. –  Bennett Gardiner Dec 4 '12 at 7:20
Try the constant solution $u(x,t) = 1$. –  Robert Israel Dec 4 '12 at 8:30
:) Unfortunately, I need something a little more complicated. –  Bennett Gardiner Dec 4 '12 at 8:56
Also, I had a mistake in the B/C, thankyou. –  Bennett Gardiner Dec 4 '12 at 8:57