Stability estimates for nonlinear hyperbolic problems with nonlinear Wentzell boundary conditions

Of concern is the nonlinear hyperbolic problem with nonlinear dynamic boundary conditions $$left{ begin{array}{lll} u_{tt} ={rm div} (mathcal{A} nabla u)-gamma (x,u_t), && quad {rm in} ; (0, infty) times Omega, u(0, cdot)=f, , u_t(0,cdot)=g, && quad {rm in}; Omega, u_{tt} + beta partial^ mathcal{A}_nu u+c(x)u+ delta (x,u_t)-q beta Lambda_{rm LB} u=0,&& quad {rm on} ;(0, infty ) times partial Omega . end{array}right. $$ for t ≥  0 and ${x in Omega subset mathbb{R}^N}$ ; the last equation holds on the boundary ?Ω. Here ${mathcal{A}= {a_{ij}(x)}_{ij}}$ is a real, hermitian, uniformly positive definite N × N matrix; ${beta in C(partial Omega)}$ , with β > 0; ${gamma:Omega times mathbb{R} to mathbb{R}; delta:partial Omega times mathbb{R} to mathbb{R}; ,c:partial Omega to mathbb{R}; , q ge 0, Lambda_{rm LB}}$ is the Laplace–Beltrami operator on ?Ω, and ${partial^mathcal{A}_nu u}$ is the conormal derivative of u with respect to ${mathcal{A}}$ ; everything is sufficiently regular. We prove explicit stability estimates of the solution u with respect to the coefficients ${mathcal{A},,beta,,gamma,,delta,,c,,q}$ , 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.