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

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• 作者：Tohru Wakasa^a ; ^{wakasa@mns.kyutech.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Shoji Yotsutani^b ; ^{shoji@math.ryukoku.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34A05 ; 34B05 ; 34B15 ; 35K57
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 November 2016
• 年：2016
• 卷：261
• 期：10
• 页码：5465-5498
• 全文大小：638 K

文摘

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 rescaled spectral problems on the whole line.