The local well-posedness and inviscid limit for a general form of Keller-Segel equation with logistic sources

In this paper, we consider the qualitative analysis for a general form of n-dimension Keller-Segel system with logistic sources (include the parabolic-elliptic Keller-Segel (PEKS) system and the corresponding hyperbolic-elliptic Keller-Segel (HEKS) system). By the transport (-diffusion) theory, we first establish the local existence and uniqueness of strong solutions to (PEKS) and (HEKS) for the initial data in B-p,r(s)(R-n) with s>max{(n)/(p),12}, 1 <= p,r <=infinity (or s=(n)/(p), 1 <= p <= 2n,r=1) and also obtain the continuity of the solution map with respect to the initial data in the space C([0;T];B-p,r(s)'(R-n))boolean AND C1([0;T];B-p,r(s)'-1(R-n)) for every s '<s when r=+infinity or s '=s when r<+infinity and then derive a continuation criterion result for (HEKS). In addition, we prove that this data-to-solution map for (PEKS) is discontinuous in the metric of B-2,infinity(s). Furthermore, we show that the inviscid limit of the (PEKS) converges to the (HEKS) in the same topology of Besov spaces as the initial data u(0)is an element of B-p,r(s)(R-n).