# zbMATH — the first resource for mathematics

On a nonlinear wave equation with boundary damping. (English) Zbl 1292.35049
Summary: This paper is concerned with the existence and decay of solutions of the mixed problem for the nonlinear wave equation $u''-\mu\Delta u+\alpha f\biggl(\int_\Omega u^2dx\biggl)u+\beta g\biggl(\int_\Omega u^{\prime^2}dx\biggl)u'=0\quad\text{in}\quad\Omega\times(0,\infty)$ with boundary conditions $$u=0\text{ on }\Gamma_0\times(0,\infty)\text{ and }\frac{\partial u}{\partial\nu}+h(\cdot,u')=0\text{ on }\Gamma_1\times (0,\infty)$$. Here, $$\Omega$$ is an open bounded set of $$\mathbb R^n$$ with boundary $$\Gamma$$ of class $$C^{2}$$; $$\Gamma$$ is constituted of two disjoint closed parts $$\Gamma _{0}$$ and $$\Gamma _{1}$$ both with positive measure; the functions $$\mu (t)$$, $$f(s)$$, $$g(s)$$ satisfy the conditions $$\mu (t)\geq \mu _{0}>0$$, $$f(s)\geq 0$$, $$g(s)\geq 0$$ for $$t\geq 0$$, $$s\geq 0$$ and $$h(x,s)$$ is a real function where $$x\in\Gamma_{1}$$, $$s\in\mathbb R$$; $$\nu (x)$$ is the unit outward normal vector at $$x\in\Gamma_1$$ and $$\alpha$$, $$\beta$$ are non-negative real constants.
Assuming that $$h(x,s)$$ is strongly monotone in $$s$$ for each $$x\in \Gamma _{1}$$, 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.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35L20 Initial-boundary value problems for second-order hyperbolic equations 35L71 Second-order semilinear hyperbolic equations 35R09 Integral partial differential equations
Full Text:
##### References:
 [1] Schiff, Nonlinear meson theory of nuclear forces. I. (Neutral scalar mesons with point-contact repulsion), The Physical Reviews. Second Series 84 (1) pp 1– (1951) · Zbl 0054.08305 [2] Jörgens, Des aufangswert in grossen für eine klasse nichtlinearer wallengleinchungen, Mathematische Zeitschrift 77 pp 295– (1961) · Zbl 0111.09105 · doi:10.1007/BF01180181 [3] Segal, Nonlinear partial differential equations in quantum field theory, Proceedings of Symposia in Applied Mathematics. American Mathematical Society 17 pp 210– (1965) · Zbl 0152.23902 · doi:10.1090/psapm/017/0202406 [4] Lions, Quelques Méthodes de Résolutions des Problèmes aux Limites Non-Linéaires (1969) [5] Zurita JG Existence and asymptotic behaviour of solutions of a nonlinear hyperbolic equation Thesis 1998 [6] Komornik, A direct method for the boundary stabilization of the wave equation, Journal de Mathematiques Pures et Appliquees 69 pp 33– (1990) · Zbl 0636.93064 [7] Milla Miranda, On a boundary value problem for wave equations: existence, uniqueness-asymptotic behavior, Revista de Matemáticas Aplicadas, Universidad de Chile 17 pp 47– (1996) · Zbl 0859.35070 [8] Araruna, Existence and boundary stabilization of the semilinear wave equation, Nonlinear Analysis 67 pp 1288– (2007) · Zbl 1151.35050 · doi:10.1016/j.na.2006.07.015 [9] Milla Miranda, Existence and boundary stabilization of solutions for the Kirchhoff equation, Communications in Partial Differential Equations 24 pp 1759– (1999) · Zbl 0930.35110 · doi:10.1080/03605309908821482 [10] Louredo, Local solutions for a coupled system of Kirchhoff type, Nonlinear Analysis 74 pp 7094– (2011) · Zbl 1230.35055 · doi:10.1016/j.na.2011.07.030 [11] Louredo, Nonlinear boundary dissipation for a coupled system of Klein-Gordon Equations, Electronic Journal of Differential Equations 120 pp 1– (2010) · Zbl 1198.35146 [12] Louredo, Boundary stabilization for a coupled system, Nonlinear Analysis 74 pp 6988– (2011) · Zbl 1230.35015 · doi:10.1016/j.na.2011.07.019 [13] Araújo, Vibrations of beams by torsion or impact, Matemática Contemporânea 36 pp 29– (2009) · Zbl 1195.35045 [14] Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM Journal on Control and Optimization 28 pp 466– (1990) · Zbl 0695.93090 · doi:10.1137/0328025 [15] Lasiecka, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations 6 pp 507– (1993) · Zbl 0819.35098 [16] Komornik, Exact Controllability and Stabilization. The Multiplier Method (1994) · Zbl 0937.93003 [17] Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, Journal of Differential Equations 186 pp 259– (2002) · Zbl 1016.35048 · doi:10.1016/S0022-0396(02)00023-2 [18] Cavalcanti, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, Journal of Differential Equations 203 pp 119– (2004) · Zbl 1049.35047 · doi:10.1016/j.jde.2004.04.011 [19] Jeong, On uniform decay of the solution for a damped nonlinear coupled system of wave equations with nonlinear boundary damping and memory term, Applied Mathematics and Computation 148 pp 207– (2004) · Zbl 1039.35060 · doi:10.1016/S0096-3003(02)00838-X [20] Jeong, On uniform decay of solutions for wave equation of Kirchhoff type with nonlinear boundary damping and memory source term, Applied Mathematics and Computation 138 pp 463– (2003) · Zbl 1024.35075 · doi:10.1016/S0096-3003(02)00163-7 [21] Strauss, On weak solutions of semilinear hyperbolic equations, Anais da Academia Brasileira de Ciências 42 pp 645– (1970) · Zbl 0217.13104 [22] Lions, Équations aux, Dérivées Partielles - Interpolation I (2003) [23] Medeiros, Espaços de Sobolev (Iniciação aos Problemas Eliticos Não Homogêneos), 5. ed. (2006) [24] Milla Miranda, Traço para o dual dos espaços de Sobolev, Boletim da Sociedade Paranaense de Matemática. (2a Séri) 11 (2) pp 131– (1990) [25] Lions, Problèmes aux Limites Non Homogènes et Applications 1 (1968) [26] Simon, Compact sets in the space Lp(0,T; B), Annali di Matematica Pura ed Applicata. Série IV CXLVI pp 65– (1987) [27] Brezis, Nonlinear Evolution Equations (1994) [28] Marcus, Every superposition operator mapping one Sobolev space into another is continuous, Journal of Functional Analysis 33 pp 217– (1979) · Zbl 0418.46024 · doi:10.1016/0022-1236(79)90113-7 [29] Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Mathematische Annalen 203 pp 403– (1996) · Zbl 0856.35084 · doi:10.1007/BF01444231
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.