Stability estimates for nonlinear hyperbolic problems with nonlinear Wentzell boundary conditions

Of concern is the nonlinear hyperbolic problem with nonlinear dynamic boundary conditions, for t ≥ 0 x ∈ Ω ⊂ ℝN; 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, ΔLB is the Laplace-Beltrami operator on ∂Ω and ∂vAu 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. © 2012 Springer Basel AG