Stability of transition front solutions in Cahn–Hilliard systems

We consider the asymptotic stability of transition front solutions for Cahn–Hilliard systems on R. Such equations arise naturally in the study of phase separation, and systems describe cases in which three or more phases are possible. When a Cahn–Hilliard system is linearized about a transition front solution, the linearized operator has an eigenvalue at 0 (due to shift invariance), which is not separated from essential spectrum. In many cases it is possible to show that the only non-negative eigenvalue is λ = 0, and so stability depends entirely on the nature of this neutral eigenvalue. In such cases, we identify a stability condition based on an appropriate Evans function, and we verify this condition under strong structural conditions on our equations. Due to lack of a spectral gap, nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate. One of our main results asserts that spectral stability implies nonlinear stability.