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=Δu-\nabla\cdot(χu\nabla v)+\nabla\cdot(ξu\nabla w), &x\in Ω, ~~t>0, v_t=D_1Δv+αu-βv,& x\in Ω, ~~t>0, w_t=D_2Δw+γu-δw, &x\in Ω, ~~t>0,\\ u(x,0)=u_0(x),~v(x,0)= v_0(x), w(x,0)= w_0(x) & x\in Ω, \end{cases} \end{equation} in a bounded domain $\Omega \subset {\mathbb{R}}^2$ 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 $\frac{\xi \gamma }{\chi \alpha }\ge \max \{\frac{D_1}{D_2},\frac{D_2}{D_1},\frac{\beta }{\delta },\frac{\delta }{\beta }\}$ 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 ${\bar{u}}_0=\frac{1}{|\Omega |}{\int }_{\Omega }{u}_{0}dx$. Furthermore, the decay rate is exponential if $\frac{\xi \gamma }{\chi \alpha }>\max \{\frac{\beta }{\delta },\frac{\delta }{\beta }\}$. 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.

20 pages