Decomposition of global solutions for a class of nonlinear wave equations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916648915566592 |
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| author | Mavrogiannis, Georgios Soffer, Avy Wu, Xiaoxu |
| author_facet | Mavrogiannis, Georgios Soffer, Avy Wu, Xiaoxu |
| contents | In the present paper we consider global solutions of a class of non-linear wave equations of the form \begin{equation*}
\Box u= N(x,t,u)u, \end{equation*} where the nonlinearity~$ N(x,t,u)u$ is assumed to satisfy appropriate boundedness assumptions.
Under these appropriate assumptions we prove that the free channel wave operator exists. Moreover, if the interaction term~$N(x,t,u)u$ is localised, then we prove that the global solution of the full nonlinear equation can be decomposed into a `free' part and a `localised' part.
The present work can be seen as an extension of the scattering results of~\cite{SW20221} for the Schrödinger equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05272 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decomposition of global solutions for a class of nonlinear wave equations Mavrogiannis, Georgios Soffer, Avy Wu, Xiaoxu Analysis of PDEs Mathematical Physics 35Q55 In the present paper we consider global solutions of a class of non-linear wave equations of the form \begin{equation*} \Box u= N(x,t,u)u, \end{equation*} where the nonlinearity~$ N(x,t,u)u$ is assumed to satisfy appropriate boundedness assumptions. Under these appropriate assumptions we prove that the free channel wave operator exists. Moreover, if the interaction term~$N(x,t,u)u$ is localised, then we prove that the global solution of the full nonlinear equation can be decomposed into a `free' part and a `localised' part. The present work can be seen as an extension of the scattering results of~\cite{SW20221} for the Schrödinger equation. |
| title | Decomposition of global solutions for a class of nonlinear wave equations |
| topic | Analysis of PDEs Mathematical Physics 35Q55 |
| url | https://arxiv.org/abs/2409.05272 |