Boundedness in a two-species quasi-linear chemotaxis system with two chemicals

Abstract

We consider the two-species quasi-linear chemotaxis system generalizing the prototype \begin{equation} \begin{cases} u_t=\nabla\cdot(D_1(u)\nabla u)-\chi_1\nabla\cdot(S_1(u)\nabla v), &x\in \Omega,\ t>0,\\ 0=\Delta v- v +w, &x\in \Omega,\ t>0,\\ w_t=\nabla\cdot(D_2(w)\nabla w)-\chi_2\nabla\cdot(S_2(w)\nabla z), & x\in \Omega,\ t>0,\\ {0=\Delta z- z +u}, & x\in \Omega,\ t>0, \end{cases} \tag*{$(0.1)$} \end{equation} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subseteq \mathbb{R}^N$ $(N\geq1)$. Here $D_i(u)=(u+1)^{m_i-1}$, $S_i(u)=u(u+1)^{q_i-1}$ $(i=1,2)$, with parameters $m_i\geq1$, $q_i>0$ and $\chi_1,\chi_2\in \mathbb{R}$. Hence, (0.1) allows the interaction of attraction-repulsion, with attraction-attraction and repulsion-repulsion type. It is proved that (i) in the attraction-repulsion case $\chi_1<0$: if $q_1<m_1+{2}/{N}$ and $q_2<m_2+{2}/{N}-{(N-2)^+}/{N}$, then for any nonnegative smooth initial data, there exists a unique global classical solution which is bounded; (ii) in the doubly repulsive case $\chi_1= \chi_2 < 0$: if $q_1<m_1+{2}/{N}-{(N-2)^+}/{N}$ and $q_2<m_2+{2}/{N}-{(N-2)^+}/{N}$, then for any nonnegative smooth initial data, there exists a unique global classical solution which is bounded; (iii) in the attraction-attraction case $\chi_1= \chi_2 >0$: if $q_1<{2}/{N}+m_1-1$ and $q_2<{2}/{N}+m_2-1$, then for any nonnegative smooth initial data, there exists a unique global classical solution which is bounded. In particular, these results demonstrate that the circular chemotaxis mechanism underlying (0.1) goes along with essentially the same destabilizing features as known for the quasi-linear chemotaxis system in the doubly attractive case. These results generalize the results of Tao and Winkler (Discrete Contin. Dyn. Syst. Ser. B. 20 (9) (2015), 3165-3183) and also enlarge the parameter range $q>{2}/{N}-1$ (see Cieślak and Winkler (Nonlinearity 21 (2008), 1057-1076)).