Asymptotic formulas for $L^2$ bifurcation curves of nonlocal logistic equation of population dynamics

TL;DR

The paper analyzes the asymptotic shape of $L^2$-bifurcation curves for a one-dimensional nonlocal Kirchhoff-type logistic equation. It employs a scaling reduction $u_\alpha = h_\alpha w_{\xi_\alpha}$ to construct solutions for large $\alpha$, leverages known asymptotics of the auxiliary problem $-w''+w^p=\gamma w$, and derives precise expansions that reveal how the nonlocal term $\|u'\|_2$ modifies the bifurcation curve. The main result is the explicit asymptotic formula $\lambda(\alpha)=\alpha^{p-1}\{1 + C_1\alpha^{-{p-1-q(p+1)}/2} + o(\alpha^{-{p-1-q(p+1)}/2})\}$, with a sharper rate in the special case $q=(p-1)/(2p)$. These findings illuminate the impact of nonlocality on $L^2$-bifurcation structure and provide a rigorous existence-uniqueness framework for large-amplitude solutions in this class of problems.

Abstract

The one-dimensional nonlocal Kirchhoff type bifurcation problems which are derived from logistic equation of population dynamics are studied. We obtain the precise asymptotic shapes of $L^2$ bifurcation curves $λ= λ(α)$ as $α\to \infty$, where $α:= \Vert u_λ\Vert_2$.