Diffusion with square of function inside time derivative I came across the following PDE for the first time today:
$$ \frac{\partial}{\partial t} (u^2(x,t))=\frac{\partial^2 u}{\partial x^2} $$
And with the help of Wolfram Alpha, I found its solution to be $u(x,t) = -\frac{c_1^2 \tanh(c_1 x +c_2 t +c_3)}{c_2}$. I checked that indeed this furnishes a solution but I was wondering a) is the solution unique given boundary / initial conditions and b) how do we find it in the first place?

It's solvable with separation of variables. Let $u(x,y) = X(x)T(t)$ then
$$ 2X^2 T\ T' = X''T $$
or
$$ 2T' = \frac{X''}{X^2} = \lambda $$
The $X$ equation is non-linear but can be transformed into a seperable ODE
$$ X'' X' = \lambda X^2 X' $$
$$ (X')^2 = \frac{2\lambda}{3} X^3 + \mu $$
