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Global classical solutions of semilinear wave equations on \({\mathbb{R}^3} \times \mathbb{T}\) with cubic nonlinearities

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Abstract

In this paper, we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space \({\mathbb{R}^3} \times \mathbb{T}\). The semilinear nonlinearity is assumed to be of the cubic form. The main ingredient here is the establishment of the L2–L∞ decay estimates and the energy estimates for the linear problem, which are adapted to the wave equation on the product space. The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction, the scaling technique, and the combination of the decay estimates and the energy estimates.

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Acknowledgements

The author is deeply indebted to his advisor Professor Huicheng YIN for the guidance and encouragement.

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  1. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Fei Tao

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  1. Fei Tao
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Tao, F. Global classical solutions of semilinear wave equations on \({\mathbb{R}^3} \times \mathbb{T}\) with cubic nonlinearities. Acta Math Sci 44, 115–128 (2024). https://doi.org/10.1007/s10473-024-0105-3

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  • DOI: https://doi.org/10.1007/s10473-024-0105-3

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