Electron. J. Differential Equations, Vol. 2022 (2022), No. 59, pp. 1-18.

Stabilization of the critical nonlinear Klein-Gordon equation with variable coefficients on R3

Song-Ren Fu, Zhen-Hu Ning

We prove the exponential stability of the defocusing critical semilinear wave equation with variable coefficients and locally distributed damping on R3. 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 5, 2022.
Math Subject Classifications: 93B05, 93C20, 35G16, 35L72, 35L15.
Key Words: Critical semilinear wave equation; variable coefficients; stability; Morawetz estimates; Riemannian geometry; unique continuation.

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Song-Ren Fu
Key Laboratory of Systems and Control
Institute of Systems Science
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing, 100190, China
email: songrenfu@amss.ac.cn
Zhen-Hu Ning
Faculty of Information Technology
Beijing University of Technology
Beijing, 100124, China
email: nzh41034@163.com

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