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Precise Spectral Asymptotics for the Dirichlet Problem − u″(t) + g(u(t)) = λsinu(t)

✍ Scribed by Tetsutaro Shibata

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 163 KB
Volume: 267
Category: Article
ISSN: 0022-247X
DOI: 10.1006/jmaa.2001.7793

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✦ Synopsis

where λ > 0 is a parameter and T > 0 is a constant. It is known that if λ 1, then the corresponding solution has boundary layers. In this paper, we characterize λ by the boundary layers of the solution when λ 1 from a variational point of view. To this end, we parameterize a solution pair λ u by a new parameter 0 < < T , which characterizes the boundary layers of the solution, and establish precise asymptotic formulas for λ with exact second term as → 0. It turns out that the second term is a constant which is explicitly determined by the nonlinearity g.

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Precise Spectral Asymptotics for the Dirichlet Problem −&#xa0;u″(t)&#xa0;+&#xa0;g(u(t))&#xa0;=&#xa0;λsinu(t)

✦ Synopsis

Precise Spectral Asymptotics for the Dirichlet Problem − u″(t) + g(u(t)) = λsinu(t)