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

TL;DR

This paper proves the global existence, boundedness, and exponential convergence to steady state of solutions for the full attraction-repulsion Keller-Segel system in two dimensions, considering unequal chemical diffusion rates.

Contribution

It provides the first analysis of the full ARKS system with unequal diffusion rates in multiple dimensions, establishing conditions for global stability.

Findings

Solutions are globally bounded under certain parameter conditions.

Solutions converge to a constant steady state as time approaches infinity.

Exponential decay rate is established under stronger parameter conditions.

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.