Asymptotic stability analysis for transition front solutions in Cahn-Hilliard systems
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- Asymptotic stability analysis for transition front solutions in Cahn-Hilliard systems
- Howard, Peter; Kwon, Bongsuk
- Asymptotic behaviors; Cahn-Hilliard systems; Eigen-value; Essential spectrum; Evans function; Linearized operators; Non-linear stabilities; Semi-group; Shift invariance; Transition fronts
- Issue Date
- ELSEVIER SCIENCE BV
- PHYSICA D-NONLINEAR PHENOMENA, v.241, no.14, pp.1193 - 1222
- 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, 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 spectral stability - a necessary condition for stability, defined in terms of an appropriate Evans function - implies nonlinear stability.
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