Problem: Find the solution behavior to$u \partial_{xt} u - \partial_x u \partial_t u + t u \partial_x u + (1 + t^2) \sin(2x) u^2 = 0, u(0, t) = e^{-t}, u_t(x, 0) = \cos^2(x)$ in the limit $t \to \infty$Please place your final solution in a $\boxed{}$ LaTeX Environment.
Solution: $\boxed{e^{\cos(x)^2 t - \log(1 + t^2)}}$