We consider the following chemotaxis system: \(\begin{aligned} {\left\{ \begin{array}{ll} u_{t}= \Delta u - \nabla \cdot (u\nabla f(v)), \; x \in \Omega \text {, }t>0,\\ v_{t}=\Delta v - u g(v), \; x \in \Omega \text {, }t>0, \end{array}\right. } \end{aligned}\) under homogeneous Neumann boundary conditions in a bounded smooth domain \(\Omega \subset \mathbb {R}^{n}, n=2,3 \) with nonlinear functions f and g. We establish the existence of a global classical solution under the smallness assumption on initial data. This result generalizes the existing findings for the minimal case, where \(f(s)=s\) and \(g(s)=s.\) We further present blow-up criteria for the system.
