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

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...

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Bibliographic Details
Published in:	Electronic journal of differential equations Vol. 2008; no. 161; pp. 1 - 7
Main Author:	Tetsutaro Shibata
Format:	Journal Article
Language:	English
Published:	Texas State University 09.12.2008
Subjects:	L^infty$-norm of solutions Logistic equation
ISSN:	1072-6691
Online Access:	Get full text
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Summary:	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 $lambda o infty$ when $p=2$ and $p=5$.
ISSN:	1072-6691