Asymptotic Lp stability for transition fronts in Cahn–Hilliard systems

We consider the asymptotic behavior of perturbations of transition front solutions arising in Cahn–Hilliard systems on R. Such equations arise naturally in the study of phase separation processes, 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 cases such as this, nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate. Our main result asserts that if initial perturbations are small in L1∩L∞ then spectral stability—a necessary condition for stability, defined in terms of an appropriate Evans function—implies asymptotic nonlinear stability in Lp for all 1