Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems
Tohru Wakasa - Department of Applied Mathematics, Waseda University, Ohkubo, Shinjuku-ku, Tokyo 169-8555, 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.
Received: January 2009; Revised: October 2009; Available Online: December 2009.
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