Author:
Wang Chang-Jian,Zhu Jia-Yue
Abstract
<abstract><p>In this paper, we study the following quasilinear chemotaxis system</p>
<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi \nabla \cdot (\varphi (u)\nabla v)-\xi \nabla \cdot (\psi(u)\nabla w)+f(u), \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta v-v+v_{1}^{\gamma_{1}}, \ 0 = \Delta v_{1}-v_{1}+u^{\gamma_{2}}, \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta w-w+w_{1}^{\gamma_{3}}, \ 0 = \Delta w_{1}-w_{1}+u^{\gamma_{4}}, \ &\ \ x\in \Omega, \ t>0, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p>
<p>in a smoothly bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1) $ with homogeneous Neumann boundary conditions, where $ \varphi(\varrho)\leq\varrho(\varrho+1)^{\theta-1}, $ $ \psi(\varrho)\leq\varrho(\varrho+1)^{l-1} $ and $ f(\varrho)\leq a \varrho-b\varrho^{s} $ for all $ \varrho\geq0, $ and the parameters satisfy $ a, b, \chi, \xi, \gamma_{2}, \gamma_{4} > 0, $ $ s > 1, $ $ \gamma_{1}, \gamma_{3}\geq1 $ and $ \theta, l\in \mathbb{R}. $ It has been proven that if $ s \geq\max\{ \gamma_{1}\gamma_{2}+\theta, \gamma_{3}\gamma_{4}+l\}, $ then the system has a nonnegative classical solution that is globally bounded. The boundedness condition obtained in this paper relies only on the power exponents of the system, which is independent of the coefficients of the system and space dimension $ n. $ In this work, we generalize the results established by previous researchers.</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)