Problem: Find an approximate solution describing the behavior for the non-linear partial differential equation $$ \frac{\partial u}{\partial t} = u^4 \frac{\partial^2 u}{\partial x^2} + \tan(u);\quad \frac{\partial u}{\partial x}(0, t) = 0,\quad u(x\to\pm\infty, t) = 0.$$. The initial condition is $$ u(x, 0) = \begin{cases} 1 & \text{if } |x| < 2 \\ 0 & \text{if } |x| \ge 2 \end{cases} $$ If the solution blows up in finite time $t^*$, seek the behavior for the limit $t \rightarrow t^*$ around the blowup point. A local approximation is sufficient. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \boxed{} LaTeX Environment. If there is any free parameter, set it equal to 5. $t^*$ should not be in the box; if there is a $t^*$, replace it with 0.5. The solution should not be in cases and should not contain any text. If there are multiple solutions please separate them with a ;. If you cannot find a solution, return zero. If you encounter a function like $\Theta$, replace it with $Theta$ with no backslash.
Solution: $$ \boxed{u(x,t) \approx \frac{\pi}{2} - \sqrt{2(0.1-t)} - 5 \frac{x^2}{\sqrt{0.1-t}}} $$