Well-posedness, regularity, asymptotic behavior and analyticity for some plate-membrane type transmission problems
In this thesis some plate-membrane type transmission problems are studied. Three dampings are considered on the structure: thermal and structural for the plate, and global viscoelastic of Kelvin-Voigt type on the membrane. Sometimes some damping is removed from the structure. The plate may or may no...
- Autores:
-
González Ospino, Jonathan
- Tipo de recurso:
- Doctoral thesis
- Fecha de publicación:
- 2022
- Institución:
- Universidad del Norte
- Repositorio:
- Repositorio Uninorte
- Idioma:
- eng
- OAI Identifier:
- oai:manglar.uninorte.edu.co:10584/11710
- Acceso en línea:
- http://hdl.handle.net/10584/11710
- Palabra clave:
- Ecuaciones integrales
- Rights
- openAccess
- License
- https://creativecommons.org/licenses/by/4.0/
| Summary: | In this thesis some plate-membrane type transmission problems are studied. Three dampings are considered on the structure: thermal and structural for the plate, and global viscoelastic of Kelvin-Voigt type on the membrane. Sometimes some damping is removed from the structure. The plate may or may not have an inertial term. In the presence and/or absence of any of the elements mentioned above, we establish existence and uniqueness of solution of the system, which depends continuously on the initial data. We also obtain results of regularity, stability and analyticity. We use the semigroup approach to show the well-posedness our system. Following an idea of proof of regularity developed by Avalos and Lasiecka, we prove that if the inertial term is present or absent then the boundary and transmission conditions hold in the strong sense of the trace when the initial data are smooth enough. Then, using a general criteria of Arendt-Batty, we show the strong stability of our system when the membrane is damped and the plate is with or without rotational inertia. Employing a spectral approach, we indirectly prove exponential stability when the plate has rotational inertia and the structure is totally damped. This asymptotic behavior of the solutions is lost when we remove the viscoelastic component of the membrane. Under this situation, we impose a geometrical condition on the membrane boundary and obtain that the solutions decay polynomially with a rate of order at least 1/25 when the plate has rotational inertia and structural damping. Finally, using a well-known Liu-Zheng criterion we prove by contradiction the analyticity of the system when the membrane has Kelvin-Voigt damping and the thermoelastic plate is considered without inertial term and without structural damping. |
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