CONTINUOUS DEPENDENCE IN HYPERBOLIC PROBLEMS WITH WENTZELL BOUNDARY CONDITIONS

Let Omega be a smooth bounded domain in R-N and let Lu = Sigma(N)(j,k=1) partial derivative(xj)(a(jk)(x)partial derivative(u)(xc)), in Omega and Lu vertical bar beta(x) Sigma(N)(j,k=1) a(jk)(x)partial derivative(xj)un(k) vertical bar gamma(x)u - q beta(x) Sigma(N-1)(j,k=1) partial derivative(tau k)(b(jk)(x)partial derivative(tau j)u) = 0, on partial derivative 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 Phi(n) - (a(jk)((n)), b(jk)((n)), beta(n), gamma(n), q(n)) satisfying Phi(n) -> Phi(0) uniformly as n -> infinity, then u(n)(t) -> u(0)(t) where u(n) satisfies idu(n)/dt = s(n)(m)u(n), or d(2)u(n)/dt(2) + S(n)(2m)u(n) = 0, or d(2)u(n)/dt(2) + F(S-n)du(n)/dt + S(n)(2m)u(n) - 0, for m = 1, 2; initial conditions independent of n, and for certain nonnegative functions F. This includes Schrodinger equations, damped and undamped wave equations, and telegraph equations.