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!