Stabilization of the critical nonlinear Klein-Gordon equation with variable coefficients on \(\mathbb{R}^3\). (English) Zbl 1504.81086
Summary: 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.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35G16 | Initial-boundary value problems for linear higher-order PDEs |
| 35L72 | Second-order quasilinear hyperbolic equations |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L71 | Second-order semilinear hyperbolic equations |
| 53B21 | Methods of local Riemannian geometry |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 93D23 | Exponential stability |
Keywords:
critical semilinear wave equation; variable coefficients; stability; Morawetz estimates; Riemannian geometry; unique continuationReferences:
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