Nonlocal eigenvalue problems arising in a generalized phase-field-type system

作者：Jimbo Shuichi; Morita Yoshihisa^*

来源：Japan Journal of Industrial and Applied Mathematics, 2017, 34(2): 555-584.

DOI：10.1007/s13160-017-0254-z

• Phase-field-type system
• Spectral comparison
• Linearized eigenvalue problem
• Nonlocal eigenvalue problem
• Mini-max principle

We deal with a generalized phase-field-type system that arises as a transformed system of reaction-diffusion equations with a conservation law. We consider the stationary problem which is reduced to a scalar elliptic equation with a nonlocal term, and study the linearized eigenvalue problem. We first prove by the spectral comparison argument that the number of unstable eigenvalues for the problem coincides with the one of the linearized eigenvalue problem for the original system. We next show a limiting behavior of eigenvalues for the scalar problem as the coefficient of the nonlocal term goes to infinity.

• 出版日期2017-8