Infinite-time concentration in Aggregation--Diffusion equations with a given potential

TL;DR

This paper investigates the long-term behavior of radial solutions to an aggregation-diffusion equation with nonlinear fast diffusion and potential-driven convection, revealing a unique infinite-time blow-up phenomenon where solutions split into a steady state plus a delta mass.

Contribution

It demonstrates the existence of an infinite-time blow-up with a splitting phenomenon in aggregation-diffusion equations, linking steady states to energy minimizers and characterizing the blow-up structure.

Findings

Steady states are explicit sums of integrable functions and Dirac deltas.

Solutions exhibit infinite-time blow-up with a mass split.

Steady states minimize a relaxed energy functional.

Abstract

Typically, aggregation-diffusion is modeled by parabolic equations that combine linear or nonlinear diffusion with a Fokker-Planck convection term. Under very general suitable assumptions, we prove that radial solutions of the evolution process converge asymptotically in time towards a stationary state representing the balance between the two effects. Our parabolic system is the gradient flow of an energy functional, and in fact we show that the stationary states are minimizers of a relaxed energy. Here, we study radial solutions of an aggregation-diffusion model that combines nonlinear fast diffusion with a convection term driven by the gradient of a potential, both in balls and the whole space. We show that, depending on the exponent of fast diffusion and the potential, the steady state is given by the sum of an explicit integrable function, plus a Dirac delta at the origin containing the rest of the mass of the initial datum. Furthermore, it is a global minimizer of the relaxed energy. This splitting phenomenon is an uncommon example of blow-up in infinite time.