Electron. J. Diff. Eqns., Vol. 2008(2008), No. 161, pp. 1-7.

Asymptotic expansion formulas for the maximum of solutions to diffusive logistic equations

Tetsutaro Shibata

Abstract:
We consider the nonlinear eigenvalue problems
$$\displaylines{ -u''(t) + u(t)^p = \lambda u(t),\cr u(t) > 0, \quad t \in I := (0, 1), \quad u(0) = u(1) = 0, }$$
where $p > 1$ is a constant and $\lambda > 0$ is a parameter. This equation is well known as the original logistic equation of population dynamics when $p=2$. We establish the precise asymptotic formula for $L^\infty$-norm of the solution $u_\lambda$ as $\lambda \to \infty$ when $p=2$ and $p=5$.

Submitted May 07, 2008. Published December 9, 2008.
Math Subject Classifications: 34B15.
Key Words: Logistic equation; $L^\infty$-norm of solutions.

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Tetsutaro Shibata
Department of Applied Mathematics
Graduate School of Engineering, Hiroshima University
Higashi-Hiroshima, 739-8527, Japan
email: shibata@amath.hiroshima-u.ac.jp

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