Based on a sample of 79 local advanced merger (adv-merger) (U)LIRGs, we search for the evidence of quenching process by investigating the distributions of the star formation history indicators (EW(H$alpha$), EW(H$delta$$_A$) and D$_n(4000)$) on the NUV-r color-mass and SFR-$M_{ast}$ diagrams. The distributions of the EW(H$alpha$) and D$_n(4000)$ on the NUV-r color-mass diagram show clear trends that at a given stellar mass, galaxies with redder NUV-r colors have smaller EW(H$alpha$) and larger D$_n(4000)$. The reddest adv-merger (U)LIRGs close to the green valley have D$_n(4000)$$>1.4$ mostly. In addition, in the SFR-$M_{ast}$ diagram, as the SFR decreases, the EW(H$alpha$) decreases and the D$_n(4000)$ increases, implying that the adv-merger (U)LIRGs on the star formation main sequence have more evolved stellar populations than those above the main sequence. These results indicate that a fraction of the adv-merger (U)LIRGs have already exhibited signs of fading from the starburst phase and that the NUV-r reddest adv-merger (U)LIRGs are likely at the initial stage of post-starbursts with age of $sim 1$ Gyr, which is consistent with the gas exhausting time-scales. Therefore, our results offer additional support for the fast evolutionary track from the blue cloud to the red sequence.
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