Problem: Find a self-similar solution to $$u_t=(u^2)_{xx}$$ for $$x\in \mathbb{R}, t>0$$. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. Box your final answer.
Solution: $$\boxed{u(x,t)=t^{-1/3}\cdot\max\{1-\frac{x^2}{12t^{2/3}},0\}}$$