scholarly journals Boundedness in a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source

2015 ◽  
Vol 65 (4) ◽  
pp. 1117-1136
Author(s):  
Ji Liu ◽  
Jia-Shan Zheng
Keyword(s):  
Logistic Source ◽  
Chemotaxis System ◽  
Quasilinear Parabolic

scholarly journals A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jie Zhao
Keyword(s):  
Boundary Conditions ◽  
Positive Constant ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Logistic Source ◽  
Chemotaxis Model ◽  
Singular Sensitivity ◽  
Dynamical Properties ◽  
Chemotaxis System ◽  
Quasilinear Parabolic

<p style='text-indent:20px;'>This paper deals with the dynamical properties of the quasilinear parabolic-parabolic chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(\frac{u}{v} \nabla v)+\mu u- \mu u^{2}, \, \, \, &amp;x\in\Omega, \, \, \, t&gt;0, \\ v_{t} = \Delta v-v+u, &amp;x\in\Omega, \, \, \, t&gt;0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a convex bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^{n} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n\geq2 $\end{document}</tex-math></inline-formula>, with smooth boundary. <inline-formula><tex-math id="M3">\begin{document}$ \chi&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \mu&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ D(u) $\end{document}</tex-math></inline-formula> is supposed to satisfy the behind properties</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha&gt;0. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>It is shown that there is a positive constant <inline-formula><tex-math id="M6">\begin{document}$ m_{*} $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{equation*} \begin{split} \int_{\Omega}u\geq m_{*} \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for all <inline-formula><tex-math id="M7">\begin{document}$ t\geq0 $\end{document}</tex-math></inline-formula>. Moreover, we prove that the solution is globally bounded. Finally, it is asserted that the solution exponentially converges to the constant stationary solution <inline-formula><tex-math id="M8">\begin{document}$ (1, 1) $\end{document}</tex-math></inline-formula>.</p>

scholarly journals On a quasilinear parabolic–elliptic chemotaxis system with logistic source

2014 ◽  
Vol 256 (5) ◽  
pp. 1847-1872 ◽  
Author(s):  
Liangchen Wang ◽  
Chunlai Mu ◽  
Pan Zheng
Keyword(s):  
Logistic Source ◽  
Chemotaxis System ◽  
Quasilinear Parabolic
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