We consider a nonlocal aggregation equation with degenerate diffusion, which describes the mean-field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all steady states must be radially decreasing up to a translation, but uniqueness (for a given mass) within the radial class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/nonuniqueness criteria are determined by the power of the degenerate diffusion, with the critical power being m = 2. In the case m ≥ 2, we show that for any attractive potential the steady state is unique for a fixed mass. In the case 1 < m < 2, we construct examples of smooth attractive potentials such that there are infinitely many radially decreasing steady states of the same mass. For the uniqueness proof, we develop a novel interpolation curve between two radially decreasing densities, and the key step is to show that the interaction energy is convex along this curve for any attractive interaction potential, which is of independent interest. © 2020 Wiley Periodicals LLC.

中文翻译：

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聚集扩散方程稳态的唯一性和非唯一性

我们考虑具有简并扩散的非局部聚集方程，它描述了由非局部相互作用和局部排斥驱动的相互作用粒子的平均场极限。当相互作用势具有吸引力时，之前已知所有稳态必须径向减少直到平移，但径向类内的唯一性（对于给定质量）是开放的，除了一些特殊的相互作用势。对于一般吸引势，我们表明唯一性/非唯一性标准由退化扩散的功率决定，临界功率为m  = 2。在m  ≥ 2的情况下，我们表明对于任何吸引势，稳态是对于固定质量是唯一的。在 1 < m的情况下  < 2，我们构建了平滑吸引势的例子，使得有无限多个相同质量的径向递减稳态。为了证明唯一性，我们在两个径向递减的密度之间开发了一条新的插值曲线，关键步骤是表明对于任何有吸引力的相互作用势，相互作用能量沿着这条曲线是凸的，这是独立的兴趣。© 2020 威利期刊有限责任公司。