Global asymptotic stability of constant equilibrium point in attraction-repulsion chemotaxis model with logistic source term

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for an attraction-repulsion chemotaxis model with logistic source term of Eq. (1) in bounded convex domains \(\Omega\subset\mathbb{R}^{n},~ n\geq1\), with smooth boundary. It is shown that if the ratio \(\frac{\mu}{\chi \alpha-\xi \gamma}\) is sufficiently large, then the unique nontrivial spatially homogeneous equilibrium given by \((u_{1},u_{2},u_{3})=(1,~\frac{\alpha}{\beta},~\frac{\gamma}{\eta})\) is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data \((u_{10},u_{20},u_{30})\) such that \(u_{10}\not\equiv0\), the above problem possesses uniquely determined global classical solution \((u_{1},u_{2},u_{3})\) with \((u_{1},u_{2},u_{3})|_{t=0}=(u_{10},u_{20},u_{30})\) which satisfies \(\left\|u_{1}(\cdot,t)-1\right\|_{L^{\infty}(\Omega)}\rightarrow{0},~~
\left\|u_{2}(\cdot,t)-\frac{\alpha}{\beta}\right\|_{L^{\infty}(\Omega)}\rightarrow{0},\left\|u_{3}(\cdot,t)-\frac{\gamma}{\eta}\right\|_{L^{\infty}(\Omega)}\rightarrow{0}\,,\) \(\mathrm{as}~t\rightarrow{\infty}\).