On a nonlinear wave equation with boundary damping

This paper is concerned with the existence and decay of solutions of the mixed problem for the nonlinear wave equation with boundary conditions Here, Ω is an open bounded set of with boundary Γ of class C2; Γ is constituted of two disjoint closed parts Γ0 and Γ1 both with positive measure; the functions μ(t), f(s), g(s) satisfy the conditions μ(t) ≥ μ0 > 0, f(s) ≥ 0, g(s) ≥ 0 for t ≥ 0, s ≥ 0 and h(x,s) is a real function where x ∈ Γ1, ν(x) is the unit outward normal vector at x ∈ Γ1 and α, β are non-negative real constants. Assuming that h(x,s) is strongly monotone in s for each x ∈ Γ1, it is proved the global existence of solutions for the previous mixed problem. For that, it is used in the Galerkin method with a special basis, the compactness approach, the Strauss approximation for real functions and the trace theorem for nonsmooth functions. The exponential decay of the energy is derived by two methods: by using a Lyapunov functional and by Nakao's method. Copyright © 2013 John Wiley & Sons, Ltd.