Continuous dependence in hyperbolic problems with Wentzell boundary conditions

Let $\Omega$ be a smooth bounded domain in $\R^N$ and let \begin{equation*} Lu=\sum_{j,k=1}^N \p_{x_j}\left(a_{jk}(x)\p_{x_k} u\right), \end{equation*} in $\Omega$ and \begin{equation*} Lu+\beta(x)\sum\limits_{j,k=1}^N a_{jk}(x)\partial_{x_j} u \, n_k+\gamma (x)u-q\beta(x)\sum_{j,k=1}^{N-1}\p_{\tau_k}\left(b_{jk}(x)\p_{\tau_j}u\right)=0, \end{equation*} on $\p\Omega$ define a generalized Laplacian on $\Omega$ with a Wentzell boundary condition involving a generalized Laplace-Beltrami operator on the boundary. Under some smoothness and positivity conditions on the coefficients, this defines a nonpositive selfadjoint operator, $-S^2$, on a suitable Hilbert space. If we have a sequence of such operators $S_0,\,S_1,\,S_2,...$ with corresponding coefficients \begin{equation*} \Phi_n=\left(a_{jk}^{(n)},\, b_{jk}^{(n)},\, \beta_n,\gamma_n,\,q_n\right) \end{equation*} satisfying $\Phi_n\to\Phi_o$ uniformly as $n\to\infty$, then $u_n(t)\to u_o(t)$ where $u_n$ satisfies \begin{equation*} i\frac{du_n}{dt}=S_n^m u_n, \end{equation*} or \begin{equation*} \frac{d^2u_n}{dt^2}+S_n^{2m} u_n=0, \end{equation*} or \begin{equation*} \frac{d^2u_n}{dt^2}+F(S_n)\frac{du_n}{dt}+S_n^{2m} u_n=0, \end{equation*} for $m=1,\,2,$ initial conditions independent of $n$, and for certain nonnegative functions $F$. This includes Schr\"odinger equations, damped and undamped wave equations, and telegraph equations.