Global existence for a class of chemotaxis systems with signal-dependent motility, indirect signal production and generalized logistic source

Abstract

In this paper, we study a chemotaxis system with signal-dependent motility, indirect signal production and generalized logistic source in a smooth bounded domain. By using a priori estimates, some important inequalities and the well-known standard Alikakos–Moser iteration, we establish the global existence of the solution for such kind of chemotaxis system. More precisely, we show that if $l>\max \left\{ \frac{n}{2},1\right\}$, $\lambda \in {\mathbb {R}}$, $\mu >0$ and $\delta >0$, then for all sufficiently smooth initial data the system

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta (\gamma (v)u)+\lambda u-\mu u^l,&{}x\in \Omega ,\,t>0,\\ v_t=\Delta v-v+w,&{}x\in \Omega ,\,t>0,\\ w_t=-\delta w+u,&{}x\in \Omega ,\,t>0, \end{array}\right. } \end{aligned}$$

possesses a unique global-in-time solution. Moreover, the solution is shown to approach

$$\begin{aligned} \left( \left( \frac{\lambda _+}{\mu }\right) ^{\frac{1}{l-1}},\frac{1}{\delta }\left( \frac{\lambda _+}{\mu }\right) ^{\frac{1}{l-1}},\frac{1}{\delta }\left( \frac{\lambda _+}{\mu }\right) ^{\frac{1}{l-1}}\right) \end{aligned}$$

in the large time limit under some extra hypotheses, where $\lambda _+=\max \{\lambda ,0\}$.