Bifurcation diagrams of one-dimensional Kirchhoff type equations

2021/09/30 by Tetsutaro Shibata, Shibata, Tetsutaro · 2 citations
Engineering · Computer Science · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2109.14864

Abstract

We study the one-dimensional Kirchhoff type equation -(b + a\Vert u'\Vert²) u''(x) = λu(x)^p, x ∈ I:= (-1,1), \enskip u(x) gt; 0, \enskip x∈ I, \enskip u(± 1) = 0, where \Vert u'\Vert = (∫_I u'(x)² dx)^1/2, a > 0, b > 0, p> 0 are given constants and λ> 0 is a bifurcation parameter. We establish the exact solution u_λ(x) and complete shape of the bifurcation curves λ= λ(ξ), where ξ:= \Vert u_λ\Vert_∞. We also study the nonlinear eigenvalue problem -\Vert u'\Vert^p-1 u''(x) = μu(x)^p, x ∈ I, \enskip u(x) gt; 0, x∈ I, \enskip u(± 1) = 0, where p > 1 is a given constant and μ> 0 is an eigenvalue parameter. We establish the first eigenvalue and eigenfunction of this problem by using a simple time map method.

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