  (Abridged) We have investigated two evolutionary scenarios advanced to
explain the centrally-brightened X-ray morphology of the supernova remnant
(SNR) W44: (1) a model involving the slow thermal evaporation of clouds
engulfed by a supernova blast wave as it propagates though a clumpy
interstellar medium (ISM), and (2) a hydrodynamical simulation of a blast wave
propagating through a homogeneous ISM, including the effects of radiative
cooling. Both models can have their respective parameters tuned to approximate
the remnant's morphology. The mean temperature of the hot plasma in W44 (~0.9
keV) as determined by our nonequilibrium ionization X-ray spectral analysis
provides the essential key to discriminate between these scenarios. Based on
the size (using the well established distance of 3 kpc) and temperature of W44,
the dynamical evolution predicted by the cloud evaporation model gives an age
for the SNR of merely 6500 yr. We argue that, because this age is inconsistent
with the characteristic age (approx. 20000 yr) of the associated PSR 1853+01,
this model cannot provide the explanation for the center-filled morphology. We
favor the radiative-phase shock model since it can reproduce both the
morphology and age of W44 assuming reasonable values for the initial explosion
energy in the range 0.7E51 to 0.9E51 ergs and the ambient ISM density of
between 3 and 4 cm**-3.
