Problem: Find a self-similarity solution for $$ \partial_t u = \partial_{xx}(u^2) - \partial_{xxxx}u + (\partial_x u)^2 $$ with maximum value at $u(0,0) = 1$.
Solution: $$ \boxed{ u(x,t) = (t+K_0^2)^{-1/2} f_s\left( \frac{x}{(t+K_0^2)^{1/4}} \right) } $$