Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production
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...
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AIMS Press
2024-11-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024293 |
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| author | Ya Tian Jing Luo |
| author_facet | Ya Tian Jing Luo |
| author_sort | Ya Tian |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-3d60084d60dc4c67b4dab6f180da5dee |
| institution | Kabale University |
| issn | 2688-1594 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | AIMS Press |
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| spelling | doaj-art-3d60084d60dc4c67b4dab6f180da5dee2025-01-23T07:53:01ZengAIMS PressElectronic Research Archive2688-15942024-11-0132116301631910.3934/era.2024293Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal productionYa Tian0Jing Luo1School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, ChinaSchool of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, ChinaIn 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.https://www.aimspress.com/article/doi/10.3934/era.2024293chemotaxisglobal existencesignal-dependent motilitynonlinear indirect signal productionlogistics source |
| spellingShingle | Ya Tian Jing Luo Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production Electronic Research Archive chemotaxis global existence signal-dependent motility nonlinear indirect signal production logistics source |
| title | Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production |
| title_full | Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production |
| title_fullStr | Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production |
| title_full_unstemmed | Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production |
| title_short | Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production |
| title_sort | boundedness and large time behavior of a signal dependent motility system with nonlinear indirect signal production |
| topic | chemotaxis global existence signal-dependent motility nonlinear indirect signal production logistics source |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024293 |
| work_keys_str_mv | AT yatian boundednessandlargetimebehaviorofasignaldependentmotilitysystemwithnonlinearindirectsignalproduction AT jingluo boundednessandlargetimebehaviorofasignaldependentmotilitysystemwithnonlinearindirectsignalproduction |