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

TL;DR

This paper establishes exponential decay for inhomogeneous, anisotropic semilinear elastic wave equations with damping, introducing a new framework that bypasses traditional Carleman estimates and applies to both critical and subcritical cases.

Contribution

It provides a checkable condition for media properties and develops a novel framework using Morawetz estimates and compactness-uniqueness to analyze stability, extending beyond classical methods.

Findings

Proved exponential decay for the wave equation with damping.

Established a new framework applicable to critical and subcritical cases.

Provided checkable conditions for media inhomogeneity and anisotropy.

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.