Global behavior of the branch of positive solutions to a logistic equation of population dynamics
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- by Tetsutaro Shibata
- Proc. Amer. Math. Soc. 136 (2008), 2547-2554
- DOI: https://doi.org/10.1090/S0002-9939-08-09311-8
- Published electronically: January 24, 2008
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Abstract:
We consider the nonlinear problem arising in population dynamics: \[ -u''(t) + u(t)^p = \lambda u(t), \qquad u(t) > 0, \quad t \in I := (0, 1), \qquad u(0) = u(1) = 0, \] 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.References
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Bibliographic Information
- Tetsutaro Shibata
- Affiliation: Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan
- Received by editor(s): June 8, 2007
- Published electronically: January 24, 2008
- Communicated by: Carmen C. Chicone
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2547-2554
- MSC (2000): Primary 34B15
- DOI: https://doi.org/10.1090/S0002-9939-08-09311-8
- MathSciNet review: 2390525