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March  2010, 9(2): 539-561. doi: 10.3934/cpaa.2010.9.539

Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems

Tohru Wakasa 1, and  Shoji Yotsutani 2,

1. 

Department of Applied Mathematics, Waseda University, Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
2. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194, Japan

Received  January 2009 Revised  October 2009 Published  December 2009

We are interested in the asymptotic profiles of all eigenfunctions for 1-dimensional linearized eigenvalue problems to nonlinear boundary value problems with a diffusion coefficient $\varepsilon$. For instance, it seems that they have simple and beautiful properties for sufficiently small $\varepsilon$ in the balanced bistable nonlinearity case. As the first step to give rigorous proofs for the above case, we study the case $f(u)=\sin u$ precisely. We show that two special eigenfunctions completely control the asymptotic profiles of other eigenfunctions.
Keywords: Nonlinear eigenvalue problem, eigenfunction, linearized eigenvalue problem, asymptotic formula., linearized stability, eigenvalue, exact solution.
Mathematics Subject Classification: 34A05, 34B05, 34B15, 35K5.
Citation: Tohru Wakasa, Shoji Yotsutani. Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems. Communications on Pure and Applied Analysis, 2010, 9 (2) : 539-561. doi: 10.3934/cpaa.2010.9.539
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