Global boundedness and large time behavior in a chemotaxis system with indirect signal consumption

Abstract

In this paper, we consider the corresponding initial-boundary value system as follows:

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla v)+f(u),&x\in \Omega ,t>0,\\&v_t=\Delta v-vw,&x\in \Omega ,t>0,\\&w_t=-\delta w+u,&x\in \Omega ,t>0\\ \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^n(n\geqslant 2)$, where the parameter $\delta >0$ and $f(u)=\mu (u-u^r)$ with $\mu \ge 0,r>1$. D, S are smooth functions satisfying $D(u)\geqslant K_0(u+1)^\alpha$, $0\le S(u)\le K_1(u+1)^{\beta -1}u$ with $\alpha ,\beta \in {\mathbb {R}}$ and $K_0,K_1>0$. When $\mu =0$, $D(u)=K_0$ and $\beta \le 1$, we proved that the system has a globally bounded classical solution, provided that $n\le 2$, or $n\ge 3$ with suitably small initial data. Furthermore, when $\mu >0$, we proved that the system possesses a globally bounded solution for $\beta <\text { max }\Bigl \{\frac{\alpha +r}{2}, \frac{\frac{n+2}{n}r+\alpha -1}{2}\Bigr \}$. Moreover, when $f(u)=\mu (u-u^2)$ in the case of $\beta =\frac{\alpha +2}{2}, n\ge 4$, we obtained the global existence and bounded solution, provided that $\mu$ is properly large. Finally, we showed these solutions converge to $(1,0,\frac{1}{\delta })$ as $t\rightarrow \infty$ and the decay state is exponential for the model with sufficiently large $\mu$ by constructing suitable functionals.