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<p>The asymptotic behavior of one-dimensional nonlocal Kirchhoff type bifurcation curves is precisely studied. This problem is motivated by the nonlocal logistic equation of population dynamics investigated recently by the author. We establish the new precise asymptotic formulas for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> bifurcation curves <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\lambda = \lambda (\alpha )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\alpha := \Vert u_\lambda \Vert _2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi>α</mi> <mo>:</mo> <mo>=</mo> <mo stretchy="false">‖</mo> </mrow> <msub> <mi>u</mi> <mi>λ</mi> </msub> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Asymptotic formulas for \(L^2\) bifurcation curves of nonlocal logistic equation of population dynamics

  • 期刊论文

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  • 发表日期：

    2026年4月7日

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  • Tetsutaro Shibata

摘要

The asymptotic behavior of one-dimensional nonlocal Kirchhoff type bifurcation curves is precisely studied. This problem is motivated by the nonlocal logistic equation of population dynamics investigated recently by the author. We establish the new precise asymptotic formulas for \(L^2\) L 2 bifurcation curves \(\lambda = \lambda (\alpha )\) λ = λ ( α ) as \(\alpha \rightarrow \infty \) α → ∞ , where \(\alpha := \Vert u_\lambda \Vert _2\) α : = ‖ u λ ‖ 2 .