  Time-decaying harmonic oscillators yield dispersive estimates with weak
decay, and change the threshold power of the nonlinearity between the short and
the long range. In the non-critical case for the time-decaying harmonic
oscillator, this threshold can be characterized by polynomial nonlinearities.
However, in the critical case, it is difficult to characterize the threshold
using only polynomial terms, and thus we use logarithmic nonlinear terms.
