Global boundedness and asymptotic behavior of solutions to a three-dimensional immune chemotaxis system

In a bounded smooth domain \(\Omega \subset \mathbb {R}^3\) $\mathrm{\Omega }\subset {\mathbb{R}}^3$ and with a positive parameter \(\chi >0\) $\chi >0$ , this paper is devoted to investigating the global boundedness and asymptotic behavior of solutions to a immune chemotaxis system. By establishing the uniform boundedness of \(L^\infty \) $L^{\infty }$ -norm for immune cells and the uniform boundedness of \(L^q\) $L^q$ -norm with \(q\in (3,\infty )\) $q\in (3,\infty )$ for the chemokines gradient, it is possible to identify a smallness condition on chemosensitivity \(\chi \) $\chi$ that ensures global existence of classical solution to the model in a three-dimensional bounded domain. Moreover, to get exponential decay rate the smallness of \((\Vert M_0\Vert _{L^\infty },\Vert A_0\Vert _{L^\infty })\) $(\Vert M_0{\Vert }_{L^{\infty }},\Vert A_0{\Vert }_{L^{\infty }})$ is pivotal, we shall see that the system still admits an energy type inequality, and an analysis thereof will show that any nontrivial solution approaches exponentially fast with respect to the topology in \(L^\infty (\Omega )\) $L^{\infty }(\mathrm{\Omega })$ .