Well-posedness and inverse problems for semilinear nonlocal wave equations
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917585162862592 |
|---|---|
| author | Lin, Yi-Hsuan Tyni, Teemu Zimmermann, Philipp |
| author_facet | Lin, Yi-Hsuan Tyni, Teemu Zimmermann, Philipp |
| contents | This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form $f(x,u)$ under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension $n\in \N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05877 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedness and inverse problems for semilinear nonlocal wave equations Lin, Yi-Hsuan Tyni, Teemu Zimmermann, Philipp Analysis of PDEs Primary 35R30, secondary 26A33, 42B37 This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form $f(x,u)$ under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension $n\in \N$. |
| title | Well-posedness and inverse problems for semilinear nonlocal wave equations |
| topic | Analysis of PDEs Primary 35R30, secondary 26A33, 42B37 |
| url | https://arxiv.org/abs/2402.05877 |