This paper deals with a quasilinear parabolic-parabolic-elliptic attraction-repulsion chemotaxis system. Boundedness, stabilization and blow-up in this system of the fully parabolic and parabolic-elliptic-elliptic versions have already been proved. The purpose of this paper is to derive boundedness and stabilization in the parabolic-parabolic-elliptic version.

We consider the following reaction-diffusion equation:$$\left(\mathrm{KS}\right)\left\{\begin{array}{cccc}{u}_t=\nabla ·\left(\nabla u^m-u^{q-1}\nabla v\right),\hfill & x\in {ℝ}^N,0<t<\infty ,0=\Delta v-v+u,\hfill & x\in {ℝ}^N,0<t<\infty ,u(x,0)=u_0\left(x\right),\hfill & x\in {ℝ}^N,\hfill \end{array}\right.$$ where $N\ge 1,m>1,q\ge max\{m+\frac{2}{N},2\}$.
In [Sugiyama, Nonlinear Anal.63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)] it was shown that in the case of $q\ge max\{m+\frac{2}{N},2\}$, the above problem (KS) is solvable globally in time for “small $L^{\frac{N(q-m)}{2}}$ data”. Moreover, the decay of the solution (u,v) in $L^p\left({ℝ}^N\right)$ was proved. In this paper, we consider the case of “$q\ge max\{m+\frac{2}{N},2\}$ and small $L^{\ell }$ data” with any fixed $\ell \ge \frac{N(q-m)}{2}$ and show that (i) there exists a time global solution (u,v) of (KS) and it decays to...