Spectral comparison between the second and the fourth order equations of conservative type with non-local terms.

*(English)*Zbl 0905.35009Summary: 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 |

##### Keywords:

instability of \(n\)-layered solution; nonlocal terms; set of steady states; Cahn-Hilliard equation
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\textit{I. Ohnishi} and \textit{Y. Nishiura}, Japan J. Ind. Appl. Math. 15, No. 2, 253--262 (1998; Zbl 0905.35009)

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##### References:

[1] | N.D. Alikakos, P.W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension. J. Diff. Eqns.,90 (1991), 81–135. · Zbl 0753.35042 · doi:10.1016/0022-0396(91)90163-4 |

[2] | P.W. Bates and P. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening. Physica, D43 (1990), 335–348. · Zbl 0706.58074 · doi:10.1016/0167-2789(90)90141-B |

[3] | P.W. Bates and P.J. Xun, Metastable patterns for the Cahn-Hilliard equations (Parts I and II). Preprint (1992), to appear in J. Diff. Eqns. |

[4] | J. Carr, M. Gurtin, and M. Slemrod, Structured phase transitions on a finite interval. Arch. Rational Mech. Anal.,86 (1984), 317–351. · Zbl 0564.76075 · doi:10.1007/BF00280031 |

[5] | R. Courant and D. Hilbert, Methods of Mathematical Physics (Vol.1). Interscience Publishers, New York, 1953. · Zbl 0051.28802 |

[6] | P. Freitas, A nonlocal Sturm-Liouville eigenvalue problem. Proc. Roy. Soc. Edinb.,124A (1994), 169–188. · Zbl 0798.34033 · doi:10.1017/S0308210500029279 |

[7] | T. Kato, Perturbation Theory for Linear Operators (Second edition). Springer-Verlag, 1980. · Zbl 0435.47001 |

[8] | F.C. LarchĂ© and J.W. Cahn, Phase changes in a thin plate with non-local self-stress effects. Acta Metall. Mater.,40 (1992), 947–955. · doi:10.1016/0956-7151(92)90071-L |

[9] | Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit. Dynamics Reported 3 (New Series), Springer-Verlag, 1994, 25–103. · Zbl 0806.35079 |

[10] | Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal.,13 (1982), 555–593. · Zbl 0501.35010 · doi:10.1137/0513037 |

[11] | Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations. SIAM J. Math. Anal.,18 (1987), 1726–1770. · Zbl 0638.35010 · doi:10.1137/0518124 |

[12] | Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers. Physica, D84 (1995), 31–39. · Zbl 1194.82069 · doi:10.1016/0167-2789(95)00005-O |

[13] | J. Rubinstein and P. Sternberg, Nonlocal reaction diffusion equations and nucleation. IMA J. Appl. Math.,48 (1992), 249–264. · Zbl 0763.35051 · doi:10.1093/imamat/48.3.249 |

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