The Fermi-LAT collaboration presented the second gamma-ray burst (GRB) catalog covering its first 10 years of operations. A significant fraction of afterglow-phase light curves in this catalog cannot be explained by the closure relations of the standard synchrotron forward-shock model, suggesting that there could be an important contribution from another process. In view of the above, we derive the synchrotron self-Compton (SSC) light curves from the reverse shock in the thick- and thin-shell regime for a uniform-density medium. We show that this emission could explain the GeV flares exhibited in some LAT light curves. Additionally, we demonstrate that the passage of the forward shock synchrotron cooling break through the LAT band from jets expanding in a uniform-density environment may be responsible for the late time ($approx10^2$ s) steepening of LAT GRB afterglow light curves. As a particular case, we model the LAT light curve of GRB 160509A that exhibited a GeV flare together with a break in the long-lasting emission, and also two very high energy photons with energies of 51.9 and 41.5 GeV observed 76.5 and 242 s after the onset of the burst, respectively. Constraining the microphysical parameters and the circumburst density from the afterglow observations, we show that the GeV flare is consistent with a SSC reverse-shock model, the break in the long-lasting emission with the passage of the synchrotron cooling break through the Fermi-LAT band and the very energetic photons with SSC emission from the forward shock when the outflow carries a significant magnetic field ($R_{rm B} simeq 30$) and it decelerates in a uniform-density medium with a very low density ($n=4.554^{+1.128}_{-1.121}times 10^{-4},{rm cm^{-3}}$).
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