On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions ✩

We study the global existence of solutions of the nonlinear degenerate wave equation (ρ 0) (∗) ρ(x)y − Δy =0 inΩ ×]0,∞[, y =0 onΓ1 ×]0,∞[, ∂y ∂ν + y + f (y) + g(y ) =0 onΓ0 ×]0,∞[, y(x, 0) = y0, (√ρy )(x, 0) = (√ρy1)(x) in Ω, where y denotes the derivative of y with respect to parameter t , f (s) = C0|s|δ s and g is a nondecreasing C1 function such that k1|s|ξ+2 g(s)s k2|s|ξ+2 for some k1, k2 > 0 with 0 < δ,ξ 1/(n −2) if n 3 or δ, ξ > 0 if n = 1, 2. The existence of solutions is proved by means of the Faedo–Galerkin method. Furthermore, when ξ = 0 the uniform decay is obtained by making use of the perturbed energy method.  2003 Elsevier Science (USA). All rights reserved.