Abstract
Klein–Gordon equation is a relativistic wave equation describing the propagation of free spinless particles in the quantum field theory. In this paper, we mainly address the Klein–Gordon equation on the curved sphere manifold with probabilistic propagation mechanism. By the application of micro-local analysis and stochastic analysis, we make a classification of multiple time-dependent oscillating coefficients on the principal harmonic oscillator and investigate the corresponding regularity behavior and \(L^2\)-estimates of the solution. Furthermore, in order to demonstrate the weak optimality of the exponential type growth rate of the \(L^2\)-estimates, typical counter-examples with periodic coefficients will be constructed to show a lower bound of exponential type growth by the application of instability arguments.
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Acknowledgements
The author thanks the anonymous referees for their instructive comments. This project is supported by Natural Science Foundation of Jiangsu Province (BK 20191257). This project is also supported by Natural Science Foundation of China (NSFC 7213018).
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Lu, X. On the Klein–Gordon equation with randomized oscillating coefficients on the sphere. Z. Angew. Math. Phys. 73, 154 (2022). https://doi.org/10.1007/s00033-022-01782-0
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DOI: https://doi.org/10.1007/s00033-022-01782-0