一般MEMS模型的径向单点破裂解
Radial single point rupture solutions for a general MEMS model
摘要：
我们研究了初始值问题$$ \begin{cases} r^{-(γ-1)}\left(r^α|u'|^{β-1}u'\right)'=\frac{1}{f(u)} & \textrm{for}\ 0<r<r_0,\\ u(r)>0 & \textrm{for}\ 0<r<r_0,\\ u(0)=0, \end{cases} $$的$γ>α>β\geq 1$和$f\in C[0,\bar u)\cap C^2(0,\bar u)$，$f(0)=0$，$f(u)>0$上$(0, \bar u)$和$f$满足一定的假设，其中包括纯功率非线性的标准情况下，在微机电系统（MEMS）的研究中遇到的。我们得到了上述问题解$u^*$的存在唯一性，它在原点处接近零值的速度以及与相应正则解$u(\,\cdot\,,a)$（与$u(0,a)=a$）的交点数为$a\to 0$ .特别是，这些结果产生的唯一性的径向单点破裂的解决方案和其他定性性质的MEMS模型。同时还研究了分岔图。
英文摘要：
We study the initial value problem $$ \begin{cases} r^{-(γ-1)}\left(r^α|u'|^{β-1}u'\right)'=\frac{1}{f(u)} & \textrm{for}\ 0<r<r_0,\\ u(r)>0 & \textrm{for}\ 0<r<r_0,\\ u(0)=0, \end{cases} $$ for $γ>α>β\geq 1$ and $f\in C[0,\bar u)\cap C^2(0,\bar u)$, $f(0)=0$, $f(u)>0$ on $(0, \bar u)$ and $f$ satisfies certain assumptions which include the standard case of pure power nonlinearities encountered in the study of Micro-Electromechanical Systems (MEMS). We obtain the existence and uniqueness of a solution $u^*$ to the above problem, the rate at which it approaches the value zero at the origin and the intersection number of points with the corresponding regular solutions $u(\,\cdot\,,a)$ (with $u(0,a)=a$) as $a\to 0$. In particular, these results yield the uniqueness of a radial single point rupture solution and other qualitative properties for MEMS models. The bifurcation diagram is also investigated.
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