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We consider the motion of an incompressible shear-thickening power-law-likenon-Newtonian fluid in $R^3$ with a variable power-law index. This system ofnonlinear partial differential equations arises in mathematical models ofelectrorheological fluids. The aim of this paper is to investigate thelarge-time behaviour of the difference $u-\tilde{u}$ where $u$ is a strongsolution of the given equations with the initial data $u_0$ and $\tilde{u}$ isthe 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 isassumed 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 sufficientlylarge $t>0$, where $\gamma\in(2,\frac{5}{2})$. The proof is based on theobservation that the solution of the linear heat equation describes theasymptotic behaviour of the solutions of the electrorheological fluids well forsufficiently large time $t>0$, and the generalized Fourier splitting methodwith an iterative argument. Furthermore, it will also be discussed that theargument used in the present paper can improve the previous results for thegeneralized Newtonian fluids with a constant power-law index.

Upper and lower bounds of convergence rates for strong solutions of the generalized Newtonian fluids with non-standard growth conditions