Hadamard Well-Posedness for two Nonlinear Structure Acoustic Models

Abstract

This dissertation focuses on the Hadamard well-posedness of two nonlinear structure acoustic models, each consisting of a semilinear wave equation defined on a smooth bounded domain $\Omega\subset\mathbb{R}^3$ strongly coupled with a Berger plate equation acting only on a flat portion of the boundary of $\Omega$. In each case, the PDE is of the following form: \begin{align*} \begin{cases} u_{tt}-\Delta u +g_1(u_t)=f(u) &\text{ in } \Omega \times (0,T),\\[1mm] w_{tt}+\Delta^2w+g_2(w_t)+u_t|_{\Gamma}=h(w)&\text{ in }\Gamma\times(0,T),\\[1mm] u=0&\text{ on }\Gamma_0\times(0,T),\\[1mm] \partial_\nu u=w_t&\text{ on }\Gamma\times(0,T),\\[1mm] w=\partial_{\nu_\Gamma}w=0&\text{ on }\partial\Gamma\times(0,T),\\[1mm] (u(0),u_t(0))=(u_0,u_1),\hspace{5mm}(w(0),w_t(0))=(w_0,w_1), \end{cases} \end{align*} where the initial data reside in the finite energy space, i.e., $$(u_0, u_1)\in H^1_{\Gamma_0}(\Omega) \times L^2(\Omega) \, \text{ and }(w_0, w_1)\in H^2_0(\Gamma)\times L^2(\Gamma).$$ The chief assumption of the first model is in taking $f(u)=-u|u|^{p-1}$, i.e., $f$ is a restoring source, where $p\geq 1$ is arbitrary. A standard Galerkin approximation scheme is used to establish a rigorous proof of the existence of local weak solutions. In addition, under some conditions on the parameters in the system, it is shown that such solutions exist globally in time and depend continuously on the initial data. For the second model, $f$ is taken to be an energy building source, and in particular it is allowed to have a \emph{supercritical} exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from $H^1_{\Gamma_0}(\Omega)$ into $L^2(\Omega)$. By employing nonlinear semigroups and the theory of monotone operators, several results on the existence of local and global weak solutions are obtained. Moreover, it is proven that such solutions depend continuously on the initial data, and uniqueness is obtained in two different scenarios. Adviser: Mohammad A. Rammah