Stabilization of the critical and subcritical semilinear inhomogeneous and anisotropic elastic wave equation

Abstract

{\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.