Global stabilization of the full attraction-repulsion Keller-Segel system

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system\begin{document}$\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \nabla \cdot (\chi u\nabla v) + \nabla \cdot (\xi u\nabla w),}&{x \in \Omega ,t > 0,}\\{{v_t} = {D_1}\Delta v + \alpha u - \beta v,}&{x \in \Omega ,t > 0,}\\{{w_t} = {D_2}\Delta w + \gamma u - \delta w,}&{x \in \Omega ,t > 0,}\\{u(x,0) = {u_0}(x),v(x,0) = {v_0}(x),w(x,0) = {w_0}(x)}&{x \in \Omega ,}\end{array}} \right.\;\;\;\;\left( * \right)$\end{document}in a bounded domain $ \Omega\subset \mathbb{R}^2 $ with smooth boundary subject to homogeneous Neumann boundary conditions. By constructing an appropriate Lyapunov functions, we establish the boundedness and asymptotical behavior of solutions to the system (*) with large initial data $ (u_0,v_0,w_0) \in [W^{1,\infty}(\Omega)]^3 $. Precisely, we show that if the parameters satisfy $ \frac{\xi\gamma}{\chi\alpha}\geq \max\Big\{\frac{D_1}{D_2},\frac{D_2}{D_1},\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\} $ for all positive parameters $ D_1,D_2,\chi,\xi,\alpha,\beta,\gamma $ and $ \delta $, the system (*) has a unique global classical solution $ (u,v,w) $, which converges to the constant steady state $ (\bar{u}_0,\frac{\alpha}{\beta}\bar{u}_0,\frac{\gamma}{\delta}\bar{u}_0) $ as $ t\to+\infty $, where $ \bar{u}_0 = \frac{1}{|\Omega|}\int_\Omega u_0dx $. Furthermore, the decay rate is exponential if $ \frac{\xi\gamma}{\chi\alpha}> \max\Big\{\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\} $. This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. $ D_1\ne D_2 $) in multi-dimensions.