Limiting classification on linearized eigenvalue problems for 1-dimensional Allen-Cahn equation II - Asymptotic profiles of eigenfunctions

作者：Wakasa Tohru^*; Yotsutani Shoji

来源：Journal of Differential Equations, 2016, 261(10): 5465-5498.

DOI：10.1016/j.jde.2016.08.016

• Reaction-diffusion equations
• Linearized eigenvalue problems
• Asymptotic formulas of eigenvalues
• Asymptotic formulas of eigenfunctions
• Floquet theory

This paper is a continuation of a previous paper by the authors. We are interested in the asymptotic behavior of eigenpairs on one dimensional linearized eigenvalue problem for Allen-Cahn equations as the diffusion coefficient tends to zero. We obtain the asymptotic profiles of all eigenfunctions by using the asymptotic formulas of corresponding eigenvalues, which have been obtained in the previous paper. Our results lead us to the concept of the classification of limiting eigenfunctions. In the case of Allen-Cahn equation it is provided by three special eigenfunctions, which correspond to the solutions of resealed spectral problems on the whole line.

• 出版日期2016-11-15