Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911401095725056 |
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| author | Perrin, Thomas |
| author_facet | Perrin, Thomas |
| contents | We study long-time dynamics of the damped focusing cubic Klein-Gordon equation on a compact three-dimensional Riemannian manifold, together with its space-independent reduction, the damped focusing Duffing equation. Under the geometric control condition on the damping and an assumption on the set of stationary solutions, we establish a sharp trichotomy for initial data with energy slightly above that of the ground state: every solution either blows up in finite time, decays exponentially to zero, or converges to a ground state. We provide a complete classification of the Duffing dynamics above the energy of the constant solution, use it to construct Klein-Gordon solutions realising each of the three behaviours in the case of a domain without boundary, and derive a simple spectral criterion ensuring that the ground states are nonconstant - and hence that different types of behaviour can indeed occur for solutions with initial energy above that of the ground state. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13851 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations Perrin, Thomas Analysis of PDEs Classical Analysis and ODEs We study long-time dynamics of the damped focusing cubic Klein-Gordon equation on a compact three-dimensional Riemannian manifold, together with its space-independent reduction, the damped focusing Duffing equation. Under the geometric control condition on the damping and an assumption on the set of stationary solutions, we establish a sharp trichotomy for initial data with energy slightly above that of the ground state: every solution either blows up in finite time, decays exponentially to zero, or converges to a ground state. We provide a complete classification of the Duffing dynamics above the energy of the constant solution, use it to construct Klein-Gordon solutions realising each of the three behaviours in the case of a domain without boundary, and derive a simple spectral criterion ensuring that the ground states are nonconstant - and hence that different types of behaviour can indeed occur for solutions with initial energy above that of the ground state. |
| title | Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2511.13851 |