  Long-lived high-energy (>100MeV) emission, a common feature of most Fermi-LAT
detected gamma-ray burst, is detected up to \sim 10^2 s in the short GRB
090510. We study the origin of this long-lived high-energy emission, using
broad-band observations including X-ray and optical data. We confirm that the
late > 100 MeV, X-ray and optical emission can be naturally explained via
synchrotron emission from an adiabatic forward shock propagating into a
homogeneous ambient medium with low number density. The Klein-Nishina effects
are found to be significant, and effects due to jet spreading and magnetic
field amplification in the shock appear to be required. Under the constraints
from the low-energy observations, the adiabatic forward shock synchrotron
emission is consistent with the later-time (t>2s) high-energy emission, but
falls below the early-time (t < 2s) high energy emission. Thus we argue that an
extra high energy component is needed at early times. A standard reverse shock
origin is found to be inconsistent with this extra component. Therefore, we
attribute the early part of the high-energy emission (t< 2s) to the prompt
component, and the long-lived high energy emission (t>2s) to the adiabatic
forward shock synchrotron afterglow radiation. This avoids the requirement for
an extremely high initial Lorentz factor.
