Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every u₀ ∈ L¹ (ℝ²). The local existence time is characterized for u ₀ ∈ L¹ ∩ L^q* (ℝ²) with 1 < q* < 2. Next, we prove the finite time blow-up of strong solution under the assumption ∥u₀∥ _L1 > 8π and ∥x|²u₀∥_L1 < 1/γ·g (∥u₀∥_L1/8π), where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.