Stabilization of the critical nonlinear Klein-Gordon equation with variable coefficients on R^3

Abstract

We prove the exponential stability of the defocusing critical semilinear wave equation with variable coefficients and locally distributed damping on \(\mathbb{R}^3\). 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.