Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915784885796864 |
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| author | Huang, Hongjing Ifrim, Mihaela Tataru, Daniel |
| author_facet | Huang, Hongjing Ifrim, Mihaela Tataru, Daniel |
| contents | In this article we consider one-dimensional scalar quasilinear Klein--Gordon equations with general nonlinearities, on both $\mathbb{R}$ and $\mathbb{T}$. By employing a refined modified-energy framework of Ifrim and Tataru, we investigate long time lifespan bounds for small data solutions. Our main result asserts that solutions with small initial data of size $ε$ persist on the improved cubic timescale $|t| \lesssim ε^{-2}$ and satisfy sharp cubic energy estimates throughout this interval. We also establish difference bounds on the same time scale. In the case of $\mathbb{R}$, we are further able to use dispersion in order to extend the lifespan to $ε^{-4}$. This generalizes earlier results obtained by Delort in the semilinear case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08055 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows Huang, Hongjing Ifrim, Mihaela Tataru, Daniel Analysis of PDEs In this article we consider one-dimensional scalar quasilinear Klein--Gordon equations with general nonlinearities, on both $\mathbb{R}$ and $\mathbb{T}$. By employing a refined modified-energy framework of Ifrim and Tataru, we investigate long time lifespan bounds for small data solutions. Our main result asserts that solutions with small initial data of size $ε$ persist on the improved cubic timescale $|t| \lesssim ε^{-2}$ and satisfy sharp cubic energy estimates throughout this interval. We also establish difference bounds on the same time scale. In the case of $\mathbb{R}$, we are further able to use dispersion in order to extend the lifespan to $ε^{-4}$. This generalizes earlier results obtained by Delort in the semilinear case. |
| title | Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.08055 |