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

We study well-posedness and long-time behaviour of aggregation–diffusion equations of the form ∂ ρ ∂ t = Δ ρ m + ∇ ⋅ ( 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 a ball and then in R d , 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 α δ 0 + ρ ˆ d x with α ≥ 0 and ρ ˆ ∈ L 1 . Finally, we give instances of W ≠ 0 showing that partial mass concentration does happen in infinite time, i.e. α > 0 .