Asymptotic behavior of bifurcation curves of ODEs with oscillatory nonlinear diffusion
Asymptotic behavior of bifurcation curves of ODEs with oscillatory nonlinear diffusion
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2019-09
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通讯作者：
T. Shibata
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作者：
T. Shibata
我们考虑非线性特征值问题$[D(u(T))u(T)‘]’+lambda g(u(T))=0$，$u(T)>；0$，$t\in I：=(0，1)$，$u(0)=u(1)=0$，它来自于多孔介质类型方程。这里，$D(U)=PU^{2n}+\sin u$($n\in{N}$，$p>；0$：给定的常量)，$g(U)=u$或$g(U)=u+\sin u$。$\lambda>；0$是分支参数，它是与$\lambda$对应的解$u_\lambda$的$\α=\vert u_\lambda\vert_\infty$的连续函数，表示为$\lambda=\lambda(\α)$。由于我们的方程在扩散项中包含了振荡项，因此研究这种振荡项对分叉曲线结构的影响具有重要意义。我们证明了最简单的情形$D(U)=u^{2n}+sin u$和$g(U)=u$给出了关于$\lambda(\α)$的全局行为的最显著的现象。
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 $\lambda(\alpha)$.