Uniform in time $L^{\infty }$-estimates for nonlinear aggregation-diffusion equations

Abstract

We derive uniform in time $L^{\infty }$-bound for solutions to an aggregation-diffusion model with attractive-repulsive potentials or fully attractive potentials. We analyze two cases: either the repulsive nonlocal term dominates over the attractive part, or the diffusion term dominates over the fully attractive nonlocal part. When the attractive potential has a weaker singularity ($2-n\le B<A\le 2$), we use the classical approach by the Sobolev and Young inequalities together with differential iterative inequalities to prove that solutions have the uniform in time $L^{\infty }$-bound. When the repulsive potential has a stronger singularity ($-n<B<2-n\le A\le 2$), we show the uniform bounds by utilizing properties of fractional operators.

References in corpus (1)

• Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics