Melt viscosities of poly(p-dioxanone) (PPDO) samples having different molecular weights were studied using a controlled-strain rotational rheometer under a nitrogen atmosphere. First, PPDO's thermal stability was evaluated by recording changes in its viscosity with time. The result, that samples' viscosities decreased with time when heated, demonstrated that PPDO is thermally unstable: its degradation activation energy, obtained by using a modified MacCallum equation, was a relatively low 71.8 kJ/mol K. Next, viscoelastic information was acquired through dynamic frequency measurements, which showed a shear thinning behavior among high molecular weight PPDOs, but a Newtonian flow behavior in a low molecular weight polymer (Mw = 18 kDa). Dynamic viscosity values were transferred to steady shear viscosities according to the Cox–Merz rule, and zero shear viscosities were derived according to the Cross model with a shear thinning index of 0.80. Then flow activation energy (48 kJ/mol K) was extrapolated for PPDO melts using an Arrhenius type equation. This activation energy is independent of polymer molecular weight. A linear relationship between zero shear viscosity and molecular weight was obtained using a double-logarithmic plot with a slope of 4.0, which is near the usually observed value of 3.4 for entangled linear polymers. Finally, the rheological behaviors of PPDO polymer blends having bimodal molecular weight distributions were investigated, with the results indicating that the relationship between zero shear viscosity and low molecular weight composition fraction can be described with a Christov model.