  {\bf Abstract} \,\,We prove exponential decay of the critical and subcritical
semilinear inhomogeneous and anisotropic elastic wave equation with locally
distributed damping on bounded domain. One novelty compared to previous
results, is to give a checkable condition of the inhomogeneous and anisotropic
medias. Another novelty is to establish a framework to study the stability of
the damped semilinear inhomogeneous and anisotropic elastic wave equation,
which is hard to apply Carleman estimates to deal with. We develop the Morawetz
estimates and the compactness-uniqueness arguments for the semiliear elastic
wave equation to prove the unique continuation, observability inequality and
stabilization result.
  It is pointing that our proof is different from the classical method (See
Dehman et al.\cite{ZYY11}, Joly et al.\cite{ZYY16} and Zuazua \cite{ZYY43}),
which succeeds for the subcritical semilinear wave equation and fails for the
critical semilinear wave equation.
