Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 161, pp. 1-7.
Title: Asymptotic expansion formulas for the maximum of
solutions to diffusive logistic equations
Authors: Tetsutaro Shibata (Hiroshima Univ., Japan)
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 09, 2008.
Math Subject Classifications: 34B15.
Key Words: Logistic equation; $L^\infty$-norm of solutions.