We consider a coupled semilinear wave system posed in an inhomogeneous medium, with smooth boundary, subject to a nonlinear damping distributed around a neighborhood of the boundary according to the Geometric Control Condition. We show that the energy of the coupled system goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase-space. The approach involves refined techniques of microlocal analysis and follows ideas due to Burq and Gérard given in (Burq and Gérard (2001)).
F.Alabau, P.Cannarsa and V.Komornik, Indirect internal stabilization of weakly coupled evolution equation, J. of Evolutions Equations2 (2002), 127–150. doi:10.1007/s00028-002-8083-0.
2.
F.Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM Journal on Control and Optimization2(41) (2002), 511–541. doi:10.1137/S0363012901385368.
3.
F.Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim.51(1) (2005), 61–105. doi:10.1007/s00245.
4.
F.Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems, J. Differential Equations248(6) (2010), 1473–1517. doi:10.1016/j.jde.2009.12.005.
5.
F.Alabau-Boussouira, Z.Whang and L.Yu, A one-step optimal energy decay formula for indrectly nonlinearly damped hyperbolic system coupled by velocities, ESAIM: Control, Optimization and Calculus of Variations23 (2017), 721–749.
6.
M.Astudillo, M.M.Cavalcanti, V.N.Domingos Cavalcanti, R.Fukuoka and A.B.Pampu, Uniform decay rates estimates for the semilinear wave equation in inhomogeneous media with locally distributed nonlinear damping, Nonlinearity31 (2018), 4031–4064. doi:10.1088/1361-6544/aac75d.
7.
V.Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Bucharest, Romania, 1976.
8.
V.Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010.
9.
C.Bardos, G.Lebeau and J.Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim.30(5) (1992), 1024–1065. doi:10.1137/0330055.
10.
S.Betelú, R.Gulliver and W.Littman, Boundary control of PDEs via curvature flows: The view from the boundary. II, Appl. Math. Optim.46(2–3) (2002), 167–178, Special issue dedicated to the memory of Jacques-Louis Lions. doi:10.1007/s00245-002-0742-6.
11.
M.D.Blair, H.F.Smith and C.D.Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire26(5) (2009), 1817–1829. doi:10.1016/j.anihpc.2008.12.004.
12.
C.A.Bortot, M.M.Cavalcanti, V.N.Domingos Cavalcanti and P.Piccione, Exponential asymptotic stability for the Klein Gordon equation on non-compact Riemannian manifolds, Appl. Math. Optim.78(2) (2018), 219–265. doi:10.1007/s00245-017-9405-5.
13.
N.Burq and P.Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. (French) [A necessary and sufficient condition for the exact controllability of the wave equation], C. R. Acad. Sci. Paris Sér. I Math.325(7) (1997), 749–752. doi:10.1016/S0764-4442(97)80053-5.
14.
N.Burq and P.Gérard, Contrôle Optimal des Équations aux Dérivées Partielles, 2001. http://www.math.u-psud.fr/~burq/articles/coursX.pdf.
15.
M.M.Cavalcanti, V.N.Domingos Cavalcanti, R.Fukuoka and J.A.Soriano, Uniform stabilization of the wave equation on compact surfaces and locally distributed damping, Methods Appl. Anal.15(4) (2008), 405–425.
16.
M.M.Cavalcanti, V.N.Domingos Cavalcanti, R.Fukuoka and J.A.Soriano, Asymptotic stability of the wave equation on compact surfaces and locally distributed damping – a sharp result, Trans. Amer. Math. Soc.361(9) (2009), 4561–4580. doi:10.1090/S0002-9947-09-04763-1.
17.
M.M.Cavalcanti, V.N.Domingos Cavalcanti, R.Fukuoka and J.A.Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: A sharp result, Arch. Ration. Mech. Anal.197(3) (2010), 925–964. doi:10.1007/s00205-009-0284-z.
18.
W.Charles, A stabilization for a coupled wave system with nonlinear and arbitrary damping, Journal of Advances in Mathematics and Computer Science26(3) (2018), 1–14. doi:10.9734/JAMCS/2018/38196.
19.
I.Chueshov, M.Eller and I.Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations27(9–10) (2002), 1901–1951. doi:10.1081/PDE-120016132.
20.
B.Dehman, Stabilisation pour l’équation des ondes semilinéaire, Asymptotic Anal.27 (2001), 171–181.
21.
B.Dehman, P.Gérard and G.Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z.254(4) (2006), 729–749. doi:10.1007/s00209-006-0005-3.
22.
B.Dehman, G.Lebeau and E.Zuazua, Stabilization and control for the subcritical semilinear wave equation, Anna. Sci. Ec. Norm. Super.36 (2003), 525–551. doi:10.1016/S0012-9593(03)00021-1.
P.Gérard, Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal.141(1) (1996), 60–98. doi:10.1006/jfan.1996.0122.
25.
P.Gérard and E.Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math J.71(2) (1993), 559–607. doi:10.1215/S0012-7094-93-07122-0.
26.
A.Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations59 (1985), 145–154. doi:10.1016/0022-0396(85)90151-2.
27.
R.Joly and C.Laurent, Stabilization for the semilinear wave equation with geometric control, Analysis & PDE6(5) (2013), 1089–1119. doi:10.2140/apde.2013.6.1089.
28.
J.Jost, Riemannian Geometry and Geometric Analysis, Springer Verlag, 2008.
29.
V.V.Kapitanov, Uniform stabilization and exact controllability for a class of coupled hiperbolic systems, Comp. Appl. Math.15 (1996), 199–212.
30.
H.Koch and D.Tataru, Dispersive estimates for principally normal pseudodifferential operators, Commun. Pure Appl. Math.58(2) (2005), 217–284. doi:10.1002/cpa.20067.
31.
I.Lasiecka and D.Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations6 (1993), 507–533.
32.
C.Laurent, On stabilization and control for the critical Klein Gordon equation on 3-d compact manifolds, Journal of Functional Analysis260 (2011), 1304–1368. doi:10.1016/j.jfa.2010.10.019.
33.
G.Lebeau, Equations des ondes amorties, in: Algebraic Geometric Methods in Maths. Physics, 1996, pp. 73–109. doi:10.1007/978-94-017-0693-3_4.
34.
J.L.Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Guthier-Villars, 1969.
35.
P.Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut.12(1) (1999), 251–283.
36.
M.Nakao, Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations, Math. Z.4 (2001), 781–797. doi:10.1007/s002090100275.
37.
P.Radu, Weak solutions to the Cauchy problem of a semilinear wave equation with damping and source terms, Adv. Differential Equations10(11) (2005), 1261–1300.
38.
M.A.Rammaha and Z.Wilstein, Hadamard well-posedness for wave equation with p-Laplacian damping and supercritical sources, Adv. Differential Equations17(1–2) (2012), 105–150.
39.
J.Rauch and M.Taylor, Exponential decay of solutions to hyperbolic equationin bounded domains, Indiana Univ. Math. J.24 (1974), 79–86. doi:10.1512/iumj.1975.24.24004.
40.
J.Rauch and M.Taylor, Decay of solutions to n ondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math.28(4) (1975), 501–523. doi:10.1002/cpa.3160280405.
41.
A.Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures. Appl.71 (1992), 455–467.
42.
E.Zuazua, Exponential decay for semilinear wave equations with localized damping, Comm. Partial Differential Equations15(2) (1990), 205–235. doi:10.1080/03605309908820681.
