Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production - Ya Tian; Jing Luo (Electronic Research Archive)
In this paper, we study a chemotaxis system with nonlinear indirect signal production \begin{document}$ \left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta \left( {\gamma \left( v \right) u } \right)}+ru-\mu u^l, \quad &x\in \Omega, t>0, \\ {{v_t} = \Delta v - v + w^{\beta}}, \quad &x\in \Omega, t>0, \\ {{w_t} = - \delta w + u}, \quad &x\in \Omega, t>0, \end{array}} \right. $\end{document} under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega \subset {\mathbb{R}^n}(n\geq2) $, where the parameters $ r $, $ \mu $, $ \beta $, $ \delta > 0 $, and $ l > 1 $, the motility function $ \gamma\in C^{3}([0, \infty)) $, $ \gamma(v) > 0 $ is bounded, $ \gamma^{'}(v) < 0 $, and $ \frac{\gamma^{'}(v)}{\gamma(v)} $ is bounded. We show that if $ {\frac{l}{\beta}} > {\frac{n}{2}} $, the system has a unique global classical solution. Moreover, the solution exponentially converges to $ ((\frac{r} {\mu})^{\frac{1}{l-1}}, (\frac{1}{\delta})^{\beta}(\frac{r}{\mu})^{\frac{\beta}{l-1}}, \frac{1}{\delta}(\frac{r}{\mu})^{\frac{1}{l-1}})) $ in the large time limit under some extra hypotheses.