Asymptotic behavior of bifurcation curves of ODEs with oscillatory nonlinear diffusion

TL;DR

This paper investigates the asymptotic behavior of bifurcation curves in nonlinear ODEs with oscillatory diffusion, revealing how oscillatory terms influence the global structure of solutions.

Contribution

It provides a detailed analysis of how oscillatory diffusion terms affect the bifurcation structure in nonlinear eigenvalue problems, especially for the case D(u) = u^{2n} + sin u.

Findings

Oscillatory diffusion significantly alters bifurcation curves.

The case D(u) = u^{2n} + sin u exhibits notable global behavior.

Results highlight the impact of oscillatory terms on solution structure.

Abstract

We consider the nonlinear eigenvalue problem $[D(u(t))u(t)']' + \lambda g(u(t)) = 0$, $u(t) > 0$, $t \in I := (0,1)$, $u(0) = u(1) = 0$, which comes from the porous media type equation. Here, $D(u) = pu^{2n} + \sin u$ ($n \in \mathbb{N}$, $p > 0$: given constants), $g(u) = u$ or $g(u) = u + \sin u$. $\lambda > 0$ is a bifurcation parameter which is a continuous function of $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution $u_\lambda$ corresponding to $\lambda$, and is expressed as $\lambda = \lambda(\alpha)$. Since our equation contains oscillatory term in diffusion term, it seems significant to study how this oscillatory term gives effect to the structure of bifurcation curves $\lambda(\alpha)$. We prove that the simplest case $D(u) = u^{2n} + \sin u$ and $g(u) = u$ gives us the most significant phenomena to the global behavior of $\lambda(\alpha)$.