Submitted Paper

Inserted: 28 aug 2025
Last Updated: 28 aug 2025

Pages: 37
Year: 2025

Abstract:

We consider an evolution model with nonlinear diffusion of porous medium type in competition with a nonlocal drift term favoring mass aggregation. The distinguishing trait of the model is the choice of a nonlinear $(s,p)$ Riesz potential for describing the overall aggregation effect. We investigate radial stationary states of the dynamics, showing their relation with extremals of suitable Hardy-Littlewood-Sobolev inequalities. In the case that aggregation does not dominate over diffusion, radial stationary states also relate to global minimizers of a homogeneous free energy functional featuring the $(s,p)$ energy associated to the nonlinear potential. In the limit as the fractional parameter $s$ tends to zero, the nonlocal interaction term becomes a backward diffusion and we describe the asymptotic behavior of the stationary states.

Keywords: Aggregation-diffusion equations, nonlinear Riesz potential, stationary states

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