  We study the spectral evolution of PWNe taking into account the energy
injected when they are young. We model the evolution of the magnetic field
inside a uniformly expanding PWN. Considering time dependent injection from the
pulsar and coolings by radiative and adiabatic losses, we solve the evolution
of the particle distribution function. The model is calibrated by fitting the
calculated spectrum to the observations of the Crab Nebula at an age of a
thousand years. The spectral evolution of the Crab Nebula in our model shows
that the flux ratio of TeV gamma-rays to X-rays increases with time, which
implies that old PWNe are faint in X-rays, but not in TeV gamma-rays. The
increase of this ratio is because the magnetic field decreases with time and is
not because the X-ray emitting particles are cooled more rapidly than the TeV
gamma-ray emitting particles. Our spectral evolution model matches the observed
rate of the radio flux decrease of the Crab Nebula. This result implies that
our magnetic field evolution model is close to the reality. Finally, from the
viewpoint of the spectral evolution, only a small fraction of the injected
energy from the Crab Pulsar needs to go to the magnetic field, which is
consistent with previous studies.
