  We explore a non-stationary outer gap scenario for gamma-ray emission process
in pulsar magnetosphere. Electrons/positrons that migrate along the magnetic
field line and enter the outer gap from the outer/inner boundaries activate the
pair-creation cascade and high-energy emission process. In our model, the rate
of the particle injection at the gap boundaries is key physical quantity to
control the gap structure and properties of the gamma-ray spectrum. Our model
assumes that the injection rate is time variable and the observed gamma-ray
spectrum are superposition of the emissions from different gap structures with
different injection rates at the gap boundaries. The calculated spectrum
superposed by assuming power law distribution of the particle injection rate
can reproduce sub-exponential cut-off feature in the gamma-ray spectrum
observed by Fermi-LAT. We fit the phase-averaged spectra for 43
young/middle-age pulsars and 14 millisecond pulsars with the model. Our results
imply that (1) a larger particle injection at the gap boundaries is more
frequent for the pulsar with a larger spin down power and (2) outer gap with an
injection rate much smaller than the Goldreich-Julian value produces observe
$>10$GeV emissions. Fermi-LAT gamma-ray pulsars show that (i) the observed
gamma-ray spectrum below cut-off energy tends to be softer for the pulsar with
a higher spin down rate and (ii) the second peak is more prominent in higher
energy bands. Based on the results of the fitting, we describe possible
theoretical interpretations for these observational properties. We also briefly
discuss Crab-like millisecond pulsars that show phase-aligned radio and
gamma-ray pulses.
