A note on construction of nonnegative initial data inducing unbounded solutions to some two-dimensional Keller-Segel systems

It was shown that unbounded solutions of the Neumann initial-boundary value problem to the two-dimensional Keller-Segel system can be induced by initial data having large negative energy if the total mass Λ ∈ (4π, ∞) \ 4π · N and an example of such an initial datum was given for some transformed system and its associated energy in Horstmann-Wang (2001). In this work, we provide an alternative construction of nonnegative nonradially symmetric initial data enforcing unbounded solutions to the original Keller-Segel model.