Boundedness in a two-species chemotaxis-consumption system with nonlinear diffusion and sensitivity

ABSTRACT

This paper deals with the following two-species chemotaxis system $\begin{array}{rl}& u_t=\mathrm{\nabla }\cdot \left(D_1\right(u\left)\mathrm{\nabla }u\right)-\mathrm{\nabla }\cdot \left(S_1\right(u\left)\mathrm{\nabla }w\right)+{\mu }_{1}u(1-u-a_{1}v),x\in \mathrm{\Omega },t>0,\\ & v_t=\mathrm{\nabla }\cdot \left(D_2\right(v\left)\mathrm{\nabla }v\right)-\mathrm{\nabla }\cdot \left(S_2\right(v\left)\mathrm{\nabla }w\right)+{\mu }_{2}v(1-a_{2}u-v),x\in \mathrm{\Omega },t>0,\\ & w_t=\mathrm{\Delta }w-(u+v)w,x\in \mathrm{\Omega },t>0\end{array}$ with homogeneous Neumann boundary conditions and suitable initial conditions, where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ($n\ge 1$) is a bounded and smooth domain. Moreover, the parameters ${\mu }_1,{\mu }_2>0$, and $a_1,a_2>0$. For i = 1, 2, we assume that $D_i\left(s\right)\ge c_{D_i}(s+1{)}^{m_i-1}\mathrm{a}\mathrm{n}\mathrm{d}{S}_i(s)\le c_{S_i}(s+1{)}^{q_i-1}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{S}_i\left(s\right)\ge 0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}s\ge 0,$ where $c_{D_i}>0$, $c_{S_i}>0$, $m_i\in \mathbb{R}$ and $q_i\in \mathbb{R}$. Then if one of the following cases holds:

1. $q_i<\frac{m_i}{2}+2-\frac{n}{n+2}$;

2. $q_i<\frac{m_i}{2}+\frac{3}{2}$ and ${\mu }_i$ is sufficiently large,

it is proved that the system possesses a unique global bounded classical solution.

Keywords:

• Two-species chemotaxis system
• boundedness
• global existence

MSC (2010) Subject Classifications:

• 92C17
• 35K55
• 35B30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

X. Hu is supported by Natural Science Foundation of Chongqing [grant No. cstc2017jcyjBX0037].