Suppose we solve Allen-Cahn on the interval $[-1,1]$
$$\epsilon^2 u_{xx} = u(u^2 - 1)$$ $$u(-1) = 0, \qquad u(1) = 0$$
For small $\epsilon$, such a solution is unique and can be chosen to be positive on the interior. Now suppose we also have a solution to the linearized equation with different boundary conditions
$$\epsilon^2\dot{u}_{xx} = (3 u^2 - 1) \dot{u}$$ $$\dot{u}(-1) = -1, \qquad \dot{u}(1) = 0$$
Can we find the sign of $\dot{u}_x(-1)$? i.e.
$$\text{sign}(\dot{u}_x(-1))$$