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

Abstract

This paper is devoted to the chemotaxis model with indirect production and general kinetic function

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u),\qquad&x\in \Omega ,\,t>0,\\&v_t=\Delta v-v+w,\qquad&x\in \Omega ,\,t>0,\\&\tau w_t+\lambda w=g(u),\qquad&x\in \Omega ,\,t>0, \end{aligned} \right. \end{aligned}$$

in a bounded domain $\Omega \subset \mathbb {R}^n(n\le 3)$ with smooth boundary $\partial \Omega$, where $\chi , \tau , \lambda$ are given positive parameters, f and g are known functions. We find several explicit conditions involving the kinetic function f, g, the parameters $\chi$, $\lambda$, and the initial data $\Vert u_0\Vert _{L^1(\Omega )}$ to ensure the global-in-time existence and uniform boundedness for the corresponding 2D/3D Neumann initial-boundary value problem. Particularly, when $f\equiv 0$, and g is a linear function, the global bounded classical solutions to the corresponding 2D Neumann initial-boundary value problem with arbitrarily large initial data and chemotactic sensitivity are established. Our results partially extend the results of Hu and Tao (Math Models Methods Appl Sci 26:2111–2128, 2016), Tao and Winkler (J Eur Math Soc 19:3641–3678, 2017), etc.