Stationary solutions for a 1D pde problem with gradient term and negative powers nonlinearity
Tóm tắt
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.
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