Equilibration of aggregation-diffusion equations with weak interaction forces

Abstract

This paper studies the large time behavior of aggregation-diffusion equations. For one spatial dimension with certain assumptions on the interaction potential, the diffusion index $m$, and the initial data, we prove the convergence to the unique steady state as time goes to infinity (equilibration), with an explicit algebraic rate. The proof is based on a uniform-in-time bound on the first moment of the density distribution, combined with an energy dissipation rate estimate. This is the first result on the equilibration of aggregation-diffusion equations for a general class of weakly confining potentials $W(r)$: those satisfying ${\lim }_{r\to \infty }W(r)<\infty$.

References in corpus (1)

• Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction

Cited by in corpus (1)

• Tightness of radially-symmetric solutions to 2D aggregation-diffusion equations with weak interaction forces