In the current work, we study a nonlocal parabolic problem with Robin boundary conditions. The problem arises from the study of an idealized electrically actuated MEMS (micro-electro-mechanical system) device, when the ends of the device are attached or pinned to a cantilever. Initially, the steady-state problem is investigated estimates of the pull-in voltage are derived. In particular, a Pohožaev's type identity is also obtained, which then facilitates the derivation of an estimate of the pull-in voltage for radially symmetric N-dimensional domains. Next a detailed study of the time-dependent problem is delivered and global-in-time as well as quenching results are obtained for generic and radially symmetric domains. The current work closes with a numerical investigation of the presented nonlocal model via an adaptive numerical method. Various numerical experiments are presented, verifying the previously derived analytical results as well as providing new insights on the qualitative behavior of the studied nonlocal model.

中文翻译：

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一个由技术引起的具有Robin边界条件的非局部问题研究

在当前的工作中，我们研究了具有 Robin 边界条件的非局部抛物线问题。该问题源于对理想化的电驱动 MEMS（微机电系统）设备的研究，当设备的末端连接或固定到悬臂时。最初，研究稳态问题，得出引入电压的估计值。特别是，还获得了 Pohožaev 的类型标识，这有助于推导出径向对称N的牵引电压估计值维域。接下来，提供了对时间相关问题的详细研究，并获得了通用域和径向对称域的全局时间和淬火结果。当前的工作以通过自适应数值方法对所提出的非局部模型进行数值研究结束。提出了各种数值实验，验证了先前导出的分析结果，并为研究的非局部模型的定性行为提供了新的见解。