Electron. J. Diff. Equ., Vol. 2013 (2013), No. 117, pp. 1-14.

Direct and inverse bifurcation problems for non-autonomous logistic equations

Tetsutaro Shibata

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.

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Tetsutaro Shibata
Laboratory of Mathematics, Institute of Engineering
Hiroshima University, Higashi-Hiroshima, 739-8527, Japan
email: shibata@amath.hiroshima-u.ac.jp

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