On a nonlinear wave equation with boundary damping

• Published in: Mathematical methods in the applied sciences 2014-06, Vol.37 (9), p.1278-1302
• Main Authors: Lourêdo, Aldo T., Ferreira de Araújo, M.A., Milla Miranda, M.
• Format: Article
• Language: English
• Subjects: Approximation; Boundaries; boundary stabilization; Decay; Functions (mathematics); Galerkin method; Galerkin methods; Mathematical analysis; Nonlinearity; special basis; Wave equations
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.2885

This paper is concerned with the existence and decay of solutions of the mixed problem for the nonlinear wave equation u′′−μ△u+αf∫Ωu2dxu+βg∫Ωu′2dxu′=0inΩ×(0,∞) with boundary conditions u=0onΓ0×(0,∞)and∂u∂ν+h(.,u′)=0onΓ1×(0,∞). Here, Ω is an open bounded set of Rn 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, s∈R; 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.