Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 117, pp. 1-14.
Title: Direct and inverse bifurcation problems for non-autonomous
logistic equations
Author: Tetsutaro Shibata (Hiroshima Univ., Higashi-Hiroshima, Japan)
Abstract:
We consider the semilinear eigenvalue problem
$$\displaylines{
-u''(t) + k(t)u(t)^p = \lambda u(t), \quad u(t) > 0, \quad
t \in I := (-1/2, 1/2), \cr
u(-1/2) = u(1/2) = 0,
}$$
where p > 1 is a constant,
and $\lambda > 0$ is a parameter. We propose a new
inverse bifurcation problem. Assume that k(t) is an unknown
function. Then can we determine k(t) from the asymptotic behavior of
the bifurcation curve?
The purpose of this paper is to answer this question affirmatively.
The key ingredient is the precise asymptotic formula for
the $L^q$-bifurcation curve $\lambda = \lambda(q,\alpha)$
as $\alpha \to \infty$ ($1 \le q < \infty$),
where $\alpha := \| k^{1/(p-1)}u_\lambda\|_q$.
Submitted December 28, 2012. Published May 10, 2013.
Math Subject Classifications: 34C23, 34B15.
Key Words: Inverse bifurcation problem; non-autonomous equation;
logistic equations.