  This paper is concerned with the attraction-repulsion chemotaxis system with
superlinear logistic degradation, \begin{align*} \begin{cases} u_t = \Delta u -
\chi \nabla\cdot(u \nabla v)
  + \xi \nabla\cdot (u \nabla w) + \lambda u - \mu u^k, \quad
  &x \in \Omega,\ t>0,\\[1.05mm] 0= \Delta v + \alpha u - \beta v, \quad
  &x \in \Omega,\ t>0,\\[1.05mm] 0= \Delta w + \gamma u - \delta w, \quad
  &x \in \Omega,\ t>0, \end{cases} \end{align*} under homogeneous Neumann
boundary conditions, in a ball $\Omega \subset \mathbb{R}^n$ ($n \ge 3$), with
constant parameters $\lambda \in \mathbb{R}$, $k>1$, $\mu, \chi, \xi, \alpha,
\beta, \gamma, \delta>0$. Blow-up phenomena in the system have been well
investigated in the case $\lambda=\mu=0$, whereas the attraction-repulsion
chemotaxis system with logistic degradation has been not studied. Under the
condition that $k>1$ is close to $1$, this paper ensures a solution which blows
up in $L^\infty$-norm and $L^\sigma$-norm with some $\sigma>1$ for some
nonnegative initial data. Moreover, a lower bound of blow-up time is derived.
