This paper is concerned with the critical threshold phenomenon for one-dimensional damped, pressureless Euler–Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J. 50 (2001) 109–157]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We also derive the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. Finally, we apply these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line.

中文翻译：

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一维阻尼欧拉-泊松系统中的临界阈值

本文关注一维阻尼、无压 Euler-Poisson 方程的临界阈值现象，该方程具有由恒定背景引起的电力，最初在 [S. Engelberg 和 H. Liu 和 E. Tadmor，印第安纳大学。数学。J. 50 (2001) 109–157]。一个简单的变换用于线性化特征方程组，这使我们能够通过相平面分析研究三种阻尼情况下临界阈值曲线的几何结构：过阻尼、欠阻尼和边界阻尼。我们还推导出这些临界曲线的显式形式。这些尖锐的结果表明，如果初始数据在阈值区域内，则解将始终保持平滑，否则将出现有限时间故障。最后，