Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Garénaux, Louis de Rijk, Björn |
| author_facet | Garénaux, Louis de Rijk, Björn |
| contents | We investigate the long-time behavior of solutions with small initial data to the viscoelastic Klein-Gordon equation with general smooth nonlinearity. Our analysis relies on the space-time resonances method to eliminate all nonresonant quadratic and cubic terms. We identify a sign condition for the remaining critical resonant term to be of absorption type, leading to global-in-time existence and diffusive decay of solutions with small initial data. Even when this condition fails, our analysis shows existence and diffusive decay of small solutions on exponentially long time intervals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02220 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation Garénaux, Louis de Rijk, Björn Analysis of PDEs 35A01, 35B40, 35L05, 74B20 We investigate the long-time behavior of solutions with small initial data to the viscoelastic Klein-Gordon equation with general smooth nonlinearity. Our analysis relies on the space-time resonances method to eliminate all nonresonant quadratic and cubic terms. We identify a sign condition for the remaining critical resonant term to be of absorption type, leading to global-in-time existence and diffusive decay of solutions with small initial data. Even when this condition fails, our analysis shows existence and diffusive decay of small solutions on exponentially long time intervals. |
| title | Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation |
| topic | Analysis of PDEs 35A01, 35B40, 35L05, 74B20 |
| url | https://arxiv.org/abs/2402.02220 |