Global existence and boundedness of a chemotaxis model with indirect production and general kinetic function (Q777341)

The author considers a chemotaxis model \[ \begin{cases} u_{t} =\Delta u+\chi \nabla \cdot (u\nabla v)+f(u), &x\in \Omega ,\ t>0, \\ v_{t} =\Delta v-v+w, &x\in \Omega ,\ t>0, \\ \tau w_{t}+\lambda w =g(u), &x\in \Omega ,\ t>0, \end{cases} \] in a bounded domain \(\Omega \) with a regular boundary, \(\Omega \subset \mathbb{R}^{n}\), \(n\leq 3\), where \(\chi ,\tau ,\lambda \) are positive parameters, and \(f\) and \(g\) are known functions. The purpose of the paper is to check if the solution to the model is global and bounded for certain general functions \(f\) and \(g\). The author provides in two main theorems explicit conditions relating the functions \(f,g\), the model parameters and the initial data, which ensure the global-in-time existence and uniform boundedness for the corresponding 2D and 3D Neumann boundary value problem. In particular, for \(f\equiv 0\) and \(g\) a linear function, the global bounded classical solution to the corresponding 2D Neumann initial-boundary value problem with arbitrarily large initial data and chemotactic sensitivity is established.