A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals
Liangchen Wang, Chunlai Mu
Abstract
<p style='text-indent:20px;'>This paper deals with the following competitive two-species chemotaxis system with two chemicals <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - {\chi _1}\nabla \cdot (u\nabla v) + {\mu _1}u\left( {1 - u - {a_1}w} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta v - v + w,}&{x \in \Omega ,t > 0,}\\{{w_t} = \Delta w - {\chi _2}\nabla \cdot (w\nabla z) + {\mu _2}w\left( {1 - w - {a_2}u} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta z - z + u,}&{x \in \Omega ,t > 0}\end{array}} \right. $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ n\geq1 $\end{document}</tex-math></inline-formula>), where the parameters <inline-formula><tex-math id="M3">\begin{document}$ \chi_i>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \mu_i>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ a_i>0 $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M6">\begin{document}$ i = 1, 2 $\end{document}</tex-math></inline-formula>). It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution if one of the following cases holds: <p style='text-indent:20px;'>(ⅰ) <inline-formula><tex-math id="M7">\begin{document}$ q_1\leq a_1; $\end{document}</tex-math></inline-formula> (ⅱ) <inline-formula><tex-math id="M8">\begin{document}$ q_2\leq a_2 $\end{document}</tex-math></inline-formula>; <p style='text-indent:20px;'>(ⅲ) <inline-formula><tex-math id="M9">\begin{document}$ q_1>a_1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ q_2> a_2 $\end{document}</tex-math></inline-formula> as well as <inline-formula><tex-math id="M11">\begin{document}$ (q_1-a_1)(q_2-a_2)<1 $\end{document}</tex-math></inline-formula>, <p style='text-indent:20px;'>where <inline-formula><tex-math id="M12">\begin{document}$ q_1: = \frac{\chi_1}{\mu_1} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ q_2: = \frac{\chi_2}{\mu_2} $\end{document}</tex-math></inline-formula>, which partially improves the results of Zhang et al. [<xref ref-type="bibr" rid="b53">53</xref>] and Tu et al. [<xref ref-type="bibr" rid="b34">34</xref>]. <p style='text-indent:20px;'>Moreover, it is proved that when <inline-formula><tex-math id="M14">\begin{document}$ a_1, a_2\in(0, 1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ \mu_1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ \mu_2 $\end{document}</tex-math></inline-formula> are sufficiently large, then any global bounded solution exponentially converges to <inline-formula><tex-math id="M17">\begin{document}$ \left(\frac{1-a_1}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_1}{1-a_1a_2}\right) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M18">\begin{document}$ t\rightarrow\infty $\end{document}</tex-math></inline-formula>; When <inline-formula><tex-math id="M19">\begin{document}$ a_1>1>a_2>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}$ \mu_2 $\end{document}</tex-math></inline-formula> is sufficiently large, then any global bounded solution exponentially converges to <inline-formula><tex-math id="M21">\begin{document}$ (0, 1, 1, 0) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M22">\begin{document}$ t\rightarrow\infty $\end{document}</tex-math></inline-formula>; When <inline-formula><tex-math id="M23">\begin{document}$ a_1 = 1>a_2>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M24">\begin{document}$ \mu_2 $\end{document}</tex-math></inline-formula> is sufficiently large, then any global bounded solution algebraically converges to <inline-formula><tex-math id="M25">\begin{document}$ (0, 1, 1, 0) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M26">\begin{document}$ t\rightarrow\infty $\end{document}</tex-math></inline-formula>. This result improves the conditions assumed in [<xref ref-type="bibr" rid="b34">34</xref>] for asymptotic behavior.