论文标题
通用MEMS模型的径向单点破裂溶液
Radial single point rupture solutions for a general MEMS model
论文作者
论文摘要
我们研究初始值问题$$ \ begin {case} r^{ - （γ-1）} \ left（r^α| u'|^{β-1} u'\ right） 0 <r <r_0，\\ u（0）= 0，\ end {cases} $$对于$γ>α>α>β\ geq 1 $和$ f \ in C [0，\ bar u）\ cap c^2（0，\ bar u）在微电机电系统（MEMS）研究中遇到的纯电源非线性。我们将解决方案$ u^*$的存在和唯一性用于上述问题，其速率与相应的常规解决方案$ u（\，\ cdot \ ,, a）$（使用$ u（0，a）= a $ a $ a \ a \ a \ 0 $ a \ 0 $ a，其接近零值的速率以及与相应的常规解决方案$ u（\，\ cdot \ ,, a）$（\，\ cdot \ ,, a）$（\，\ cdot \ ,, a）的相交数。 特别是，这些结果产生了径向单点破裂溶液和MEMS模型的其他定性特性的唯一性。还研究了分叉图。
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.
