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 ∩ Lq* (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.
All Science Journal Classification (ASJC) codes
- Mathematics (miscellaneous)