On a fully parabolic singular chemotaxis-(growth) system with indirect signal production or consumption

Abstract

This paper deals with a fully parabolic singular chemotaxis-(growth) system with indirect signal production or consumption

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

under homogeneous Neumann boundary conditions in a smooth convex bounded domain $\Omega \subset {\mathbb {R}}^{2}$, where $\chi >0$. When $f(u)=0$, $h(v,w)=-v+w$, we prove that the solution (u, v, w) of this model is globally uniformly bounded in time and exponentially converges to the steady state $(\overline{u_{0}},\overline{u_{0}},\overline{u_{0}})$ in the norm of $L^{\infty }(\Omega )$ provided that $\chi <\chi _{0}$ with $\chi _{0}>0$, where $\overline{u_{0}}:=\frac{1}{|\Omega |}\int _{\Omega }u_{0}(x)\mathrm{d}x$. Moreover, in the case of $f(u)=ru-\mu u^{2}$, $h(v,w)=-v+w$, where $r\in {\mathbb {R}}$, and $\mu >0$, we obtain the global existence of solutions for this system. Furthermore, under the conditions of $f(u)=ru-\mu u^{2}$, $h(v,w)=-vw$ and $\mu >0$, $r>0$, the solution (u, v, w) is also global in time.