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 u ′ ′ − μ △ u + αf ∫ Ω u 2 dx u + βg ∫ Ω u ′ 2 dx u ′ = 0 in Ω × ( 0 , ∞ ) with boundary conditions u = 0 onΓ 0 × ( 0 , ∞ ) and∂u ∂ν + h ( . , u ′ ) = 0 onΓ 1 × ( 0 , ∞ ) .Here, Ω is an open bounded set of R n with boundary Γ of class C 2 ; Γ is constituted of two disjoint closed parts Γ 0 and Γ 1 both with positive measure; the functions μ ( t ), f ( s ), g ( s ) satisfy the conditions μ ( t ) ≥  μ 0  > 0, f ( s ) ≥ 0, g ( s ) ≥ 0 for t  ≥ 0, s  ≥ 0 and h ( x , s ) is a real function where x  ∈ Γ 1 , s ∈ R ; ν ( x ) is the unit outward normal vector at x  ∈ Γ 1 and α , β are non‐negative real constants. Assuming that h ( x , s ) is strongly monotone in s for each x  ∈ Γ 1 , 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.