  We study the evolution of a non-relativistically expanding thin shell in
radio-emitting tidal disruption events (TDEs) based on a one-dimensional
spherically symmetric model considering the effect of both a time-dependent
mass loss rate of the disk wind and the ambient mass distribution. The
analytical solutions are derived in two extreme limits: one is the approximate
solution near the origin in the form of the Taylor series, and the other is the
asymptotic solution in which the ambient matter is dominant far away from the
origin. Our numerical solutions are confirmed to agree with the respective
analytical solutions. We find that no simple power-law of time solution exists
in early to middle times because the mass loss rate varies over time, affecting
the shell dynamics. We also discuss the application of our model to the
observed radio-emitting TDE, AT2019dsg.
