Partial mass concentration for fast-diffusions with non-local aggregation terms

Abstract

We study well-posedness and long-time behaviour of aggregation-diffusion equations of the form ∂ρ∂t=Δρm+∇⋅(ρ(∇V+∇W∗ρ)) in the fast-diffusion range, 0<m<1, and V and W regular enough. We develop a well-posedness theory, first in the ball and then in Rd, and characterise the long-time asymptotics in the space W−1,1 for radial initial data. In the radial setting and for the mass equation, viscosity solutions are used to prove partial mass concentration asymptotically as t→∞, i.e. the limit as t→∞ is of the form αδ0+ρˆdx with α≥0 and ρˆ∈L1. Finally, we give instances of W≠0 showing that partial mass concentration does happen in infinite time, i.e. α>0.