Stationary solutions for a 1D pde problem with gradient term and negative powers nonlinearity
Stationary solutions for a 1D pde problem with gradient term and negative powers nonlinearity
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具有梯度项和负幂非线性的一维偏微分方程问题的平稳解
DOI：
10.1007/s41808-022-00180-x
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发表时间：
2022
影响因子：
0.8
通讯作者：
Sakamoto Takashi Okuda
中科院分区：
文献类型：
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作者：
Ichida Yu;Sakamoto Takashi Okuda
研究了一类带梯度项和负幂非线性项的一维偏微分方程的平稳解。该方程是以MEMS器件的现象为背景的MEMS方程。然而，从非线性项的影响来理解解的行为并不容易。因此，本文的目的是研究一个平稳的解决方案，这是一个典型的解决方案的性质。也就是说，我们证明了存在的平稳解决方案，包括奇异性，并提供有关他们的形状和渐近行为的信息。这里，具有奇异性的稳态解意味着允许无穷大的解或具有无穷大微分系数的解。这些研究通过应用框架，结合庞加莱-李雅普诺夫紧化和经典动力系统理论。运用这些方法的关键是揭示常微分方程定常解所满足的包含无穷大的动力学性质。
Stationary solutions for the one-dimensional partial differential equation with gradient term and negative powers nonlinearity are considered. This equation is a kind of MEMS equation that has the phenomena of MEMS (micro-electro mechanical system) devices as its background. However, it is not easy to understand the behavior of the solution from the effect of the nonlinear term. Therefore, the purpose of this paper is to investigate the properties of a stationary solution that is a typical solution. That is, we prove the existence of stationary solutions including singularities, and give information about their shapes and the asymptotic behavior. Here, the stationary solution with singularity here means a solution that allows infinity or a solution with an infinite differential coefficient. These are studied by applying the framework that combines the Poincaré–Lyapunov compactification and classical dynamical systems theory. The key to use these methods is to reveal the dynamics including infinity of an ordinary differential equation satisfied by stationary solutions.