Abstract
We present a spectral comparison theorem between the second and the fourth order equations of conservative type with non-local terms. Nonlocal effects arise naturally due to the long-range spatial connectivity in polymer problems or to the difference of relaxation times for phase separation problems with stress effect. If such nonlocal effects are built into the usual Cahn-Hilliard dynamics, we have the fourth order equations with nonlocal terms. We introduce the second order conservative equations with the same nonlocal terms as the fourth order ones. The aim is to show that both the second and the fourth order equations have the same set of steady states and their stability properties also coincide with each other. This reduction from the fourth order to the second order is quite useful in applications. In fact a simple and new proof for the instability ofn-layered solution of the Cahn-Hilliard equation is given with the aid of this reduction.
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Dedicated to Professor Kyūya Masuda on the occasion of his 60th birthday.
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Ohnishi, I., Nishiura, Y. Spectral comparison between the second and the fourth order equations of conservative type with non-local terms. Japan J. Indust. Appl. Math. 15, 253 (1998). https://doi.org/10.1007/BF03167403
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DOI: https://doi.org/10.1007/BF03167403