Limiting classification on linearized eigenvalue problems for 1-dimensional Allen-Cahn equation I-asymptotic formulas of eigenvalues

作者：Wakasa Tohru^*; Yotsutani Shoji

来源：Journal of Differential Equations, 2015, 258(11): 3960-4006.

DOI：10.1016/j.jde.2015.01.023

• Reaction-diffusion equations
• Linearized eigenvalue problems
• Representation of eigenfunctions
• Asymptotic formulas of eigenvalues
• Elliptic integrals
• Lame equation

We are interested in classical linearized eigenvalue problems for one dimensional reaction diffusion equations with small diffusion coefficient. Previously, authors have obtained the precise asymptotic formulas of all eigenvalues and eigenfunctions for f (u) = sin u, which suggest interesting structure of eigenpairs: existence of two special eigenpairs and the limiting classification of the others by special eigenpairs. In this paper we study the Allen-Cahn case f (u) = u - u(3) and obtain the asymptotic formulas of eigenvalues, which leads us to the limiting classification of eigenvalues.

• 出版日期2015-6-5