Problem: For $0 < x < 1$, $t > 0$ and a small parameter $0 < \epsilon \ll 1$ consider the PDE $u_t = \epsilon u_{xx} + u(1-u), \quad u(0, t) = 1, u(1, t) = 1/2, u(x, 0) =1.$. Find the leading order solution as $t \to \infty$. Please place your final solution in a $\boxed{}$ LaTeX Environment
Solution: $\boxed{u(x,t \to \infty) \approx \frac{3}{2} \left(\frac{(5+2\sqrt{6}) e^{(1-x)/\sqrt{\epsilon}} - 1}{(5+2\sqrt{6}) e^{(1-x)/\sqrt{\epsilon}} + 1}\right)^2 - \frac{1}{2}}$