Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant - Jing Zhang (Electronic Journal of Qualitative Theory of Differential Equations, 2019)
This paper deals with a two-species chemotaxis system \begin{equation*} \begin{cases} u_t=\nabla\cdot(D_1(u)\nabla u)-\nabla\cdot(u\chi_1(w)\nabla w)+\mu_1 u(1-u-a_1v),\quad &x\in \Omega,\quad t>0,\\ v_t=\nabla\cdot(D_2(v)\nabla v)-\nabla\cdot(v\chi_2(w)\nabla w)+\mu_2 v(1-a_2u-v),\quad &x\in \Omega,\quad t>0,\\ w_t=\Delta w-(\alpha u+\beta v)w,\quad &x\in\Omega,\quad t>0, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^n$ ($n\geq 1$) is a bounded domain with smooth boundary $\partial\Omega$; ${\chi}_i (i=1,2)$ are chemotactic functions satisfying ${\chi}'_i\geq0$; the parameters $\mu_1, \mu_2>0, a_1, a_2>0$ and $\alpha, \beta>0$, the initial data $(u_0,v_0)\in (C^0(\overline{\Omega}))^2$ and $w_0\in W^{1,\infty}(\Omega)$ are non-negative. Based on the maximal Sobolev regularity, it is shown that this system possesses a unique global bounded classical solution provided that the logistic growth coefficients $\mu_1$ and $\mu_2$ are sufficiently large.