We show that partial mass concentration can happen for stationary solutions of aggregation–diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions N ≥ 6. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions N ≥ 3, for homogeneous interaction potentials with higher power.

中文翻译：

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快速扩散导致 Keller-Segel 型固定溶液中的部分质量浓度

我们表明，在快速扩散范围内具有均匀吸引核的聚集-扩散方程的固定解可能会发生部分质量浓度。更准确地说，我们证明了自由能在一组概率测度中允许一个径向全局最小化器，它的一部分质量可能集中在给定点的狄拉克三角洲中。在四次相互作用势的情况下，我们找到了扩散指数的确切范围，其中集中发生在空间维度上ñ ≥ 6. 然后，我们提供数值计算，表明在所有维度上都出现质量浓度ñ ≥ 3，对于具有更高功率的均匀相互作用势。