We investigate the large time behavior of multi-dimensional aggregation equations driven by Newtonian repulsion, and balanced by radial attraction and confinement. In case of Newton repulsion with radial confinement we quantify the algebraic convergence decay rate toward the unique steady state. To this end, we identify a one-parameter family of radial steady states, and prove dimension-dependent decay rate in energy and 2-Wassertein distance, using a comparison with properly selected radial steady states. We also study Newtonian repulsion and radial attraction. When the attraction potential is quadratic it is known to coincide with quadratic confinement. Here, we study the case of perturbed radial quadratic attraction, proving that it still leads to one-parameter family of unique steady states. It is expected that this family to serve for a corresponding comparison argument which yields algebraic convergence toward steady repulsive-attractive solutions.

中文翻译：

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牛顿斥力和径向约束：向稳态收敛

我们研究了由牛顿斥力驱动的多维聚合方程的大时间行为，并通过径向吸引和限制来​​平衡。在具有径向约束的牛顿斥力的情况下，我们量化代数收敛衰减率朝向独特的稳态。为此，我们识别一个径向稳态态的一个参数系列，并使用与正确选择的径向稳态的比较证明能量和2-Wassertein距离中的维度依赖性衰减率。我们还研究牛顿斥力和径向引力。当吸引势为二次方时，已知与二次限制相一致。在这里，我们研究了受扰动的径向二次引力的情况，证明它仍然会导致一个参数族的独特稳态。