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

Abstract

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system \begin{equation}\label{ARKS}\tag{$\ast$} \begin{cases} 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{cases} \end{equation} in a bounded domain $\Omega\subset \R^2$ with smooth boundary subject to homogeneous Neumann boundary conditions. %The parameters $D_1,D_2,\chi,\xi,\alpha,\beta,\gamma$ and $\delta$ are positive. By constructing an appropriate Lyapunov functions, we establish the boundedness and asymptotical behavior of solutions to the system \eqref{ARKS} with large initial data. 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 \eqref{ARKS} 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.