Continuous dependence in hyperbolic problems with Wentzell boundary conditions

Let $Omega$ be a smooth bounded domain in $R^N$ and let egin{equation*} Lu=sum_{j,k=1}^N p_{x_j}left(a_{jk}(x)p_{x_k} u ight), end{equation*} in $Omega$ and egin{equation*} Lu+eta(x)sumlimits_{j,k=1}^N a_{jk}(x)partial_{x_j} u , n_k+gamma (x)u-qeta(x)sum_{j,k=1}^{N-1}p_{ au_k}left(b_{jk}(x)p_{ au_j}u ight)=0, end{equation*} on $pOmega$ 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 egin{equation*} Phi_n=left(a_{jk}^{(n)},, b_{jk}^{(n)},, eta_n,gamma_n,,q_n ight) end{equation*} satisfying $Phi_n oPhi_o$ uniformly as $n oinfty$, then $u_n(t) o u_o(t)$ where $u_n$ satisfies egin{equation*} i rac{du_n}{dt}=S_n^m u_n, end{equation*} or egin{equation*} rac{d^2u_n}{dt^2}+S_n^{2m} u_n=0, end{equation*} or egin{equation*} rac{d^2u_n}{dt^2}+F(S_n) rac{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.