This paper investigates the Keller–Segel type system \(\begin{aligned} \left\{ \begin{array}{lll} \displaystyle u_{t}=\varepsilon \Delta u-\nabla \cdot (u\nabla v)+\gamma u-\mu u^{1+\alpha },& (x,t)\in \Omega \times \mathbb {R}^{+},\\ 0=\Delta v+u^{\beta }-v,& (x,t)\in \Omega \times \mathbb {R}^{+},\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0, & (x,t)\in \partial \Omega \times \mathbb {R}^{+},\\ \displaystyle u(x,0)=u_{0}(x),& x\in \Omega \end{array}\right. \end{aligned}\) in a bounded domain \(\Omega \subset \mathbb {R}^{N}(N\ge 1)\) with smooth boundary. Here \(\alpha ,\beta ,\gamma ,\mu >0\) , and \(\varepsilon \ge 0\) . For the hyperbolic–elliptic case ( \(\varepsilon =0\) ), we prove that if \(\beta \ge 1\) and either \(\alpha >\beta \) or \(\alpha =\beta \) with \(\mu >1\) , the system admits a unique global strong solution. Conversely, if \(\alpha <\beta \) or \(\alpha =\beta \) with \(\mu <1\) , finite time blow-up in the \(L^{\infty }\) norm occurs for initial data \(u_{0}\) with sufficiently large \(\Vert u_{0}\Vert _{L^{p}(\Omega )}\) , where \(p>\max \{N,\frac{1}{1-\mu }\}\) . Notably, these results remain valid even when \(\Omega \) is not convex, thus generalizing previous work. For the parabolic–elliptic case with \(\varepsilon >0\) , it has been demonstrated that under the conditions \(\alpha \ge \beta \) , \(\beta <1\) , and \(u_0(x)>0\) , if \(\mu >2\max \{1,\gamma ^{1-\frac{\alpha }{\beta }}\}\) , then the unique classical solution (u, v) converges to the homogeneous equilibrium: \(\begin{aligned}\lim _{t\rightarrow \infty }\Big (\Big \Vert u(\cdot ,t)-\Big (\frac{\gamma }{\mu }\Big )^{\frac{1}{\alpha }}\Big \Vert _{L^{\infty }(\Omega )}+\Big \Vert v(\cdot ,t)-\Big (\frac{\gamma }{\mu }\Big )^{\frac{\beta }{\alpha }}\Big \Vert _{L^{\infty }(\Omega )}\Big )=0.\end{aligned}\) This extends previous results, which were restricted to \(\beta \ge 1\) , to demonstrate stability for sublinear production rates \((\beta <1)\) .
