Uniqueness of weak solutions to a high dimensional Keller–Segel equation with degenerate diffusion and nonlocal aggregation

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This paper considers weak solutions to the degenerate Keller–Segel equation with nonlocal aggregation: ut=Δum−∇⋅(uB(u)) in Rd×R+, where B(u)=∇((−Δ)−β2u), d≥3, β∈[2,d), 1<m<2−βd. In a previous paper of the authors (Hong et al., 2015), a criterion was established for global existence versus finite time blow-up of weak solutions to the problem. A natural question is whether the uniqueness is true for the weak solutions obtained. A positive answer is given in this paper that the global weak solutions must be unique provided the second moment of initial data is finite, which means that the weak solutions are weak entropy solutions in fact. The framework of the proof is based on the optimal transportation method.