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

AbstractWe study the global existence of solutions of the nonlinear degenerate wave equation (ρ⩾0) (∗)ρ(x)y″−Δy=0inΩ×]0,∞[,y=0onΓ1×]0,∞[,∂y∂ν+y′+f(y)+g(y′)=0onΓ0×]0,∞[,y(x,0)=y0,(ρy′)(x,0)=(ρy1)(x)inΩ, where y′ denotes the derivative of y with respect to parameter t, f(s)=C0|s|δs 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