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

Abstract

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every $$u_0 \in L^1 (\mathbb {R}^2)$$ . The local existence time is characterized for $$u_0 \in L^1 \cup L^{q*}(\mathbb {R}^2)$$ with 1 < q * < 2. Next, we prove the finite time blow-up of strong solution under the assumption $$||u_0||_{L^{1}} > 8 \pi$$ and $$||x|^2u_0||{L^1} < \frac {1}{\gamma}.g (||u_0||{L^1}/8\pi)$$ , 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.