Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals

Abstract

<p style='text-indent:20px;'>This paper studies the quasilinear attraction-repulsion chemotaxis system of two-species with two chemicals <inline-formula><tex-math id="M1">\begin{document}$ u_{t} = \nabla\cdot( D_1(u)\nabla u)-\nabla\cdot( \Phi_1(u)\nabla v) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0 = \Delta v-v+w^{\gamma_1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ w_{t} = \nabla\cdot( D_2(w)\nabla w)+\nabla\cdot( \Phi_2(w)\nabla z) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 0 = \Delta z-z+u^{\gamma_2} $\end{document}</tex-math></inline-formula>, subject to the homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M5">\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M6">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula>) with smooth boundary, where <inline-formula><tex-math id="M7">\begin{document}$ \gamma_i>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ D_i,\Phi_i\in C^2[0,+\infty) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ D_i(s)\ge(s+1)^{p_i},\; \Phi_i(s)\ge0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M10">\begin{document}$ s\ge 0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M11">\begin{document}$ \Phi_i(s)\le\chi_i s^{q_i} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M12">\begin{document}$ s>s_0 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M13">\begin{document}$ \chi_i>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M14">\begin{document}$ p_i,q_i\in\mathbb{R} $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M15">\begin{document}$ (i = 1,2) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M16">\begin{document}$ s_0>1 $\end{document}</tex-math></inline-formula>. It is shown that if <inline-formula><tex-math id="M17">\begin{document}$ \gamma_1<\frac{2}{N} $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M18">\begin{document}$ \gamma_2<\frac{4}{N} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M19">\begin{document}$ \gamma_2\le1 $\end{document}</tex-math></inline-formula>), the global boundedness of solutions are guaranteed by the self-diffusion dominance of <inline-formula><tex-math id="M20">\begin{document}$ u $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M21">\begin{document}$ w $\end{document}</tex-math></inline-formula>) with <inline-formula><tex-math id="M22">\begin{document}$ p_1>q_1+\gamma_1-1-\frac{2}{N} $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M23">\begin{document}$ p_2>q_2+\gamma_2-1-\frac{4}{N} $\end{document}</tex-math></inline-formula>); if <inline-formula><tex-math id="M24">\begin{document}$ p_j\ge q_i+\gamma_i- 1-\frac{2}{N} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M25">\begin{document}$ i,j = 1,2 $\end{document}</tex-math></inline-formula> (i.e. the self-diffusion of <inline-formula><tex-math id="M26">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M27">\begin{document}$ w $\end{document}</tex-math></inline-formula> are dominant), then the solutions are globally bounded; in particular, different from the results of the single-species chemotaxis system, for the critical case <inline-formula><tex-math id="M28">\begin{document}$ p_j = q_i+\gamma_i- 1-\frac{2}{N} $\end{document}</tex-math></inline-formula>, the global boundedness of the solutions can be obtained.</p>