  In this paper, we mainly study large time behavior of the strong solutions to
the 2-D compressible finite extensible nonlinear elastic (FENE) dumbbell model.
The Fourier splitting method yields that the $L^2$ decay rate is
$\ln^{-l}(e+t)$ for any $l\in\mathbb{N}$. By virtue of the time weighted energy
estimate, we can improve the decay rate to $(1+t)^{-\frac{1}{4}}$. Under the
low-frequency condition and by the Littlewood-Paley theory, we show that the
solutions belong to some Besov space with negative index and obtain the optimal
$L^2$ decay rate.
