The damped focusing cubic wave equation on a bounded domain
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909134450851840 |
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| author | Perrin, Thomas |
| author_facet | Perrin, Thomas |
| contents | For the focusing cubic wave equation on a compact Riemannian manifold of dimension $3$, the dichotomy between global existence and blow-up for solutions starting below the energy of the ground state is known since the work of Payne and Sattinger. In the case of a damped equation, we prove that the dichotomy between global existence and blow-up still holds. In particular, the damping does not prevent blow-up. Assuming that the damping satisfies the geometric control condition, we then prove that any global solution converges to a stationary solution along a time sequence, and that global solutions below the energy of the ground state can be stabilised, adapting the proof of a similar result in the defocusing case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12644 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The damped focusing cubic wave equation on a bounded domain Perrin, Thomas Analysis of PDEs For the focusing cubic wave equation on a compact Riemannian manifold of dimension $3$, the dichotomy between global existence and blow-up for solutions starting below the energy of the ground state is known since the work of Payne and Sattinger. In the case of a damped equation, we prove that the dichotomy between global existence and blow-up still holds. In particular, the damping does not prevent blow-up. Assuming that the damping satisfies the geometric control condition, we then prove that any global solution converges to a stationary solution along a time sequence, and that global solutions below the energy of the ground state can be stabilised, adapting the proof of a similar result in the defocusing case. |
| title | The damped focusing cubic wave equation on a bounded domain |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.12644 |