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Spectral comparison between the second and the fourth order equations of conservative type with non-local terms. (English) Zbl 0905.35009
Summary: We present a spectral comparison theorem between the second and the fourth order equations of conservative type with nonlocal 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 of $$n$$-layered solution of the Cahn-Hilliard equation is given with the aid of this reduction.

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35B35 Stability in context of PDEs 35P05 General topics in linear spectral theory for PDEs
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