Communications on Pure and Applied Analysis (CPAA)

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

Pages: 539 - 561, Volume 9, Issue 2, March 2010      doi:10.3934/cpaa.2010.9.539

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Tohru Wakasa - Department of Applied Mathematics, Waseda University, Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan (email)
Shoji Yotsutani - Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194, Japan (email)

Abstract: 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, linearized eigenvalue problem, linearized stability, eigenvalue, eigenfunction, exact solution, asymptotic formula.
Mathematics Subject Classification:  34A05, 34B05, 34B15, 35K57.

Received: January 2009;      Revised: October 2009;      Available Online: December 2009.