Stability Estimates for Nonlinear Hyperbolic Problems with nonlinear Wentzell boundary conditions

Of concern is the nonlinear hyperbolic problem with nonlinear dynamic boundary conditions utt = div( A∇u) − γ (x, ut ), in (0, ∞) × Ω, u(0, ·) = f , ut (0, ·) = g, in Ω, utt + β∂ Aν u + c(x)u + δ(x, ut ) − qβ∆LB u = 0, on (0, ∞) × ∂ Ω. for t ≥ 0 and x ∈ Ω ⊂ R^N ; the last equation holds on the boundary ∂ Ω. Here A = {aij (x)}ij is a real, hermitian, uniformly positive definite N × N matrix; β ∈ C (∂ Ω), with β > 0; γ : Ω × R → R; δ : ∂ Ω × R → R; c : ∂ Ω → R; q ≥ 0, ∆LB is the Laplace-Beltrami operator on ∂ Ω, and ∂ Aν u is the conormal derivative of u with respect to A; everything is sufficiently regular. We prove explicit stability estimates of the solution u with respect to the coefficients A, β, γ , δ, c, q, and the initial conditions f , g. Our arguments cover the singular case of a problem with q = 0 which is approximated by problems with positive q.