Global existence versus blow-up in a high dimensional Keller–Segel equation with degenerate diffusion and nonlocal aggregation

In this paper, we study the degenerate Keller–Segel equation with nonlocal aggregation ut=Δum−∇⋅(uB(u))ut=Δum−∇⋅(uB(u)) in Rd×R+Rd×R+, where m>1m>1, d≥3d≥3, and B(u)=∇((−Δ)−β2u) with β∈[2,d)β∈[2,d). By analyzing the interaction between the degenerate diffusion and the nonlocal aggregation, we determine the conditions for initial data under which weak solutions globally exist or blow up in finite time with m∈(1,d+νd), ν=d−βν=d−β. Furthermore, a sharper criterion is given for global existence and finite time blow-up of weak solutions with mm in the subrange (2d2d−ν,d+νd)⊂(1,d+νd).