Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 59, pp. 1-18.

Title: Stabilization of the critical nonlinear Klein-Gordon equation with variable coefficients on R<sup>3</sup>
       
Authors: Song-Ren Fu  (Chinese Academy of Sciences, Beijing, China)
         Zhen-Hu Ning (Beijing Univ. of Technology, Beijing, China)

Abstract:   
We prove the exponential stability of the defocusing critical semilinear wave equation
with variable coefficients and locally distributed damping on R<sup>3</sup>.
The construction of the variable coefficients is almost equivalent
to the geometric control condition.
We develop the traditional Morawetz estimates and
the compactness-uniqueness arguments for the semilinear wave equation to prove the
unique continuation result. The observability inequality and the exponential stability
are obtained subsequently.

Submitted May 11, 2022. Published August 05, 2022.
Math Subject Classifications: 93B05, 93C20, 35G16, 35L72, 35L15.
Key Words: Critical semilinear wave equation; variable coefficients; stability;
           Morawetz estimates; Riemannian geometry; unique continuation.