  In this paper we mainly study large time behavior for the strong solutions of
the finite extensible nonlinear elastic (FENE) dumbbell model. There is a lot
results about the $L^2$ decay rate of the co-rotation model. In this paper, we
consider the general case. We prove that the optimal $L^2$ decay rate of the
velocity is $(1+t)^{-\frac{d}{4}}$ with $d\geq 2$. This result improves the
previous result in \cite{Luo-Yin}.
