Open AccessHadamard well-posedness for a structure acoustic model with a supercritical source and damping terms
Author(s) -
Andrew R. Becklin,
Mohammad A. Rammaha
Publication year - 2021
Publication title -
evolution equations and control theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 19
eISSN - 2163-2480
pISSN - 2163-2472
DOI - 10.3934/eect.2020093
Subject(s) - uniqueness , omega , lipschitz continuity , bounded function , mathematics , monotone polygon , hadamard transform , operator (biology) , combinatorics , mathematical analysis , physics , quantum mechanics , geometry , biochemistry , chemistry , repressor , transcription factor , gene
This article is concerned with Hadamard's well posedness of a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain \begin{document}$ \Omega\subset\mathbb{R}^3 $\end{document} which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of \begin{document}$ \Omega $\end{document} . The system is influenced by several competing forces. In particular, the source term acting on the wave equation is allowed to have a supercritical exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from \begin{document}$ H^1_{\Gamma_0}(\Omega) $\end{document} into \begin{document}$ L^2(\Omega) $\end{document} . By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions. Moreover, we prove that such solutions depend continuously on the initial data, and uniqueness is obtained in two different scenarios.