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 \(B_{p,r}^{s}({\mathbb {R}}^n)\) with \(s> \max \{\frac{n}{p},\frac{1}{2}\}\) , \(1\le p,r \le \infty \) (or \(s=\frac{n}{p}\) , \(1\le p\le 2n, r=1\) ) and also obtain the continuity of the solution map with respect to the initial data in the space \({\mathcal {C}}([0;T];B^{s'}_{p,r}({\mathbb {R}}^n))\cap {\mathcal {C}}^1([0;T];B^{s'-1}_{p,r}({\mathbb {R}}^n))\) for every \(s'<s\) when \(r=+\infty \) or \(s'=s\) when \(r<+\infty \) 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 \(B_{2,\infty }^s\) . 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 \(u_0\in B_{p,r}^s({\mathbb {R}}^n)\) .
