The following degenerate parabolic system modelling chemotaxis is considered:where m 1, q 2, τ = 0 or 1, and N 1. The aim of this paper is to prove the existence of a time global weak solution (u, v) of (KS) with the L ∞ (0, ∞; L ∞ (R N )) bound. Such a global bound is obtained in the case of (i) m > q − 2 N for large initial data and (ii) 1 m q − 2 N for small initial data. In the case of (ii), the decay properties of the solution (u, v) are also discussed.

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We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every u 0 ∈ L 1 (R 2 ). The local existence time is characterized for u 0 ∈ L 1 ∩ L q * (R 2 ) with 1 < q * < 2. Next, we prove the finite time blow-up of strong solution under the assumption u 0 L 1 > 8π and x| 2 u 0 L 1 < 1 γ ·g( u 0 L 1 /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.

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