  We consider the motion of an incompressible shear-thickening power-law-like
non-Newtonian fluid in $R^3$ with a variable power-law index. This system of
nonlinear partial differential equations arises in mathematical models of
electrorheological fluids. The aim of this paper is to investigate the
large-time behaviour of the difference $u-\tilde{u}$ where $u$ is a strong
solution of the given equations with the initial data $u_0$ and $\tilde{u}$ is
the strong solution of the same equations with perturbed initial data
$u_0+w_0$. The initial perturbation $w_0$ is not required to be small, but is
assumed to satisfy certain decay condition. In particular, we can show that
$(1+t)^{-\frac{\gamma}{2}}\lesssim
\|u(t)-\tilde{u}(t)\|_2\lesssim(1+t)^{-\frac{\gamma}{2}}$, for sufficiently
large $t>0$, where $\gamma\in(2,\frac{5}{2})$. The proof is based on the
observation that the solution of the linear heat equation describes the
asymptotic behaviour of the solutions of the electrorheological fluids well for
sufficiently large time $t>0$, and the generalized Fourier splitting method
with an iterative argument. Furthermore, it will also be discussed that the
argument used in the present paper can improve the previous results for the
generalized Newtonian fluids with a constant power-law index.
