  We consider the global existence and asymptotic behavior of classical
solutions to the ellipsoidal BGK model for polyatomic molecules when the
initial data starts sufficiently close to a global polyatomic Maxwellian. We
observe that the linearized relaxation operator is decomposed into a trully
polyatomic part and a essentially monatomic part, leading to a dichotomy in the
dissipative property in the sense that the degeneracy of the dissipation shows
an abrupt jump as the relaxation parameter $\theta$ reaches zero. Accordingly,
we employ two different sets of micro-macro system to derive the full
coercivity and close the energy estimate.
