Asymptotic behavior of bifurcation curves of ODEs with oscillatory nonlinear diffusion

Abstract

We consider the nonlinear eigenvalue problem $[D(u(t))u(t{)}^{\prime }{]}^{\prime }+\lambda g(u(t))=0$, $u(t)>0$, $t\in I:=(0,1)$, $u(0)=u(1)=0$, which comes from the porous media type equation. Here, $D(u)=p{u}^{2n}+\sin u$ ($n\in \mathbb{N}$, $p>0$: 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 $\lambda (\alpha )$.

15 pages