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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Global behavior of the branch of positive solutions to a logistic equation of population dynamics

Author: Tetsutaro Shibata
Journal: Proc. Amer. Math. Soc. 136 (2008), 2547-2554
MSC (2000): Primary 34B15
Published electronically: January 24, 2008
MathSciNet review: 2390525
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Abstract: We consider the nonlinear problem arising in population dynamics:

$\displaystyle -u''(t) + u(t)^p = \lambda u(t), \enskip u(t) > 0, \quad t \in I := (0, 1), \enskip u(0) = u(1) = 0, $

where $ p > 1$ is a constant and $ \lambda > 0$ is a positive parameter. We establish the crucial asymptotic formula for the branch of positive solutions $ \lambda_q(\alpha)$ in $ L^q$-framework as $ \alpha \to \infty$, where $ \alpha := \Vert u_\lambda\Vert_q$ ( $ 1 \le q < \infty$). Especially, for the original logistic equation, namely the case where $ p = 2$ and $ q = 1$, we obtain not only the asymptotic expansion formula for $ \lambda_1(\alpha)$ but also the remainder estimate. Such a formula for the bifurcation branch seems to be new.

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Additional Information

Tetsutaro Shibata
Affiliation: Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

PII: S 0002-9939(08)09311-8
Keywords: $L^q$-bifurcation branch, asymptotic formula
Received by editor(s): June 8, 2007
Published electronically: January 24, 2008
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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