Abstract
We consider the initial-boundary value problem on \(\mathbb {R}^{+}\times \mathbb {R}^{+}\) for some one dimensional systems of quasilinear wave equations with null conditions. We first prove that for homogeneous Dirichlet boundary values and sufficiently small initial data, classical solutions always globally exist. Then we show that the global solution will scatter, i.e., it will converge to some solution of one dimensional linear wave equations as time tends to infinity, in the energy sense. Finally we prove the following rigidity result: if the scattering data vanish, then the global solution will also vanish identically.
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Acknowledgements
The authors would like to express their sincere gratitude to the referees for their helpful comments and suggestions. The first author is supported by the Fundamental Research Funds for the Central Universities (No. 2232022D-27).
Funding
This study was supported by National Natural Science Foundation of China (Grant No. 11801068).
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Zha, D., Xue, X. On Initial-Boundary Value Problems for Some One Dimensional Quasilinear Wave Equations: Global Existence, Scattering and Rigidity. J Geom Anal 33, 302 (2023). https://doi.org/10.1007/s12220-023-01370-2
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DOI: https://doi.org/10.1007/s12220-023-01370-2
Keywords
- One dimensional quasilinear wave equations
- Initial-boundary value problem
- Global existence
- Scattering
- Rigidity