Existence of large-data solutions to a thermo-piezoelectric system and forward operator analysis for associated inverse problems
We consider an inverse problem governed by the initial-boundary value problem for the thermo-piezoelectric Kelvin-Voigt dynamical system \[ \left\{ \begin{aligned} \rho(z,t) u_{tt} &= \frac{d}{dz} \big( \Gamma(\Theta) u_{zt} +p_1 u_z +p_2(z,t)\phi_z^0 +p_2(z,t)\chi_z -\beta \Theta \big), \\[1ex] 0 &=-\frac{d}{dz} \big( p_2(z,t)u_z -p_3(z,t)\phi_z^0 -p_3(z,t)\chi_z\big), \\ b(z,t)\Theta_t &= \frac{d}{dz}(k(z,t)\Theta_z) +\Gamma(\Theta)u_{zt}^2 -\beta \Theta u_{zt}. \end{aligned} \right. \] in an open bounded interval $\Omega\subset\mathbb{R}$, for the evolution of the displacement variable $u$, the electric potential $\phi^0$ and the temperature $\Theta\geq 0$, where $\chi$ is a given Dirichlet lift function. Assuming that the coefficients $\beta, p_1 \in \mathbb{R}^+$ and the the parameter functions $\rho$, $\Gamma$, $p_2$, $p_3$, $b$ and $k$ are strictly positive and bounded, a global-in-time existence result is established for weak solutions. We show that this can be achieved under energy- and entropy-minimal assumptions, in the sense that global weak solutions are shown to exist for any initial data $$u_0\in W^{1,2}(\Omega)\mbox{ with }u_0|_{\partial\Omega}\in\mathbb{R},\quad u_{0t}\in L^2(\Omega)\mbox{ with }u_{0t}|_{\partial\Omega}\in\mathbb{R}\quad\mbox{and}\quad 0\leq\Theta_0\in L^2(\Omega).$$ The qualitative analysis of the evolution problem then allows to model and analyze the structural properties of the corresponding forward operator arising in inverse parameter identification. Therein, two modeling approaches of the observation operator as approximations of the electrical surface charge are presented and results on their well-definedness and boundedness are established. Building on these results, we prove well-definedness, boundedness, and continuous Fr'{e}chet differentiability of the forward operator.