Problem: Find the self similarity solution for the nonlinear partial differential equation $$\partial_t = \partial_x (u^3 \partial_xu) $$ Where the initial shape of the solution is $u(x, t)=u(x, 0)= 0.9 * \exp(-x^2 /2)$ There should be a free parameter A in the solution which you should set to $A =1.1$. Please put your final solution in a $\boxed{}$ LaTeX Environment.
Solution: $$\boxed{u(x,t) = \frac{1}{(5t)^{\frac{1}{5}}} \sqrt[3]{-\frac{3}{2}\left(\frac{x}{(5t)^{\frac{1}{5}}}\right)^2 + A}}$$