Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^3 $\end{document}</tex-math></inline-formula> which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. The system is influenced by several competing forces. In particular, the source term acting on the wave equation is allowed to have a <i>supercritical</i> exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from <inline-formula><tex-math id="M3">\begin{document}$ H^1_{\Gamma_0}(\Omega) $\end{document}</tex-math></inline-formula> into <inline-formula><tex-math id="M4">\begin{document}$ L^2(\Omega) $\end{document}</tex-math></inline-formula>. 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.</p>