Global boundedness of the immune chemotaxis system with general kinetic functions

Abstract

In this paper, we study the following reaction-diffusion-advection system

$$\begin{aligned} \left\{ \begin{aligned}{}&u_t=D_{u}\Delta u- \chi \nabla \cdot ( u\nabla v) + f(u),&(x,t)\in \Omega \times (0,\infty ), \\&v_t=D_{v}\Delta v+s_{v}uw-\mu _{v}v,&(x,t)\in \Omega \times (0,\infty ), \\&w_t=D_{w}\Delta w+s_{w}-\lambda _{w}uw-\mu _{w}w,&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$

in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^{n}$, which describes a directed movement of immune cells toward chemokines during the immune process, where $D_{u},D_{v},D_{w},s_{v},s_{w},\lambda _{w}$, $\mu _{v},\mu _{w},\chi$ are positive parameters, and $f\in C^{1}([0,\infty ))$ is a kinetic function. When $n\ge 1$, if there exist positive constants $\alpha$ and $\theta _{0}$ such that $\sup _{s\ge 0}\{f(s)\!+\!\alpha s\}<\infty$ and $\lim _{s\rightarrow \infty }\inf \left\{ -\!\frac{f(s)}{s^{2}}\right\} =:\mu \in (\theta _{0},\infty ]$, then the solution of the system is global and uniformly bounded. In particular, when $n=2$ and $f(0)\ge 0$, the condition of f(u) could be improved as follows: if there exists $\alpha >0$ such that $\sup _{s\ge 0}\{f(s)+\alpha s\}<\infty$ and one of the conditions that $\lim _{s\rightarrow \infty }\inf \left\{ -\frac{f(s)\ln s}{s^{2}}\right\} =:\mu \in (\sqrt{2}\frac{\chi s_{v}C_{w}}{D_{v}},\infty ]$ or $\frac{2\sqrt{2}\chi s_{v}C_{GN}^{4}m_{1}C_{w}}{D_{v}}\le D_{u}$ holds, then the solution of the system is still global and uniformly bounded, where $m_{1}$ is a positive constant given by below, $C_{GN}>0$ is the Gagliardo-Nirenberg inequality’s constant and $C_{w}$ represents the uniform upper bound of w. Moreover, when $f\equiv 0$ and $\frac{2\sqrt{2}\chi s_{v}C_{GN}^{4}m_{1}C_{w}}{D_{v}}\le D_{u}$, the global and uniform boundedness of solutions can also be established.