×

Stability for the mixed problem involving the wave equation, with localized damping, in unbounded domains with finite measure. (English) Zbl 1406.35052

Summary: This paper is concerned with the study of the uniform decay rates of the energy associated with mixed problems involving the wave equation with nonlinear localized damping. The domain is an unbounded open set of \(\mathbb{R}^2\) with finite measure and has an unbounded smooth boundary \(\Gamma=\Gamma_N \cup \Gamma_D\) such that \(\overline{\Gamma}_N\cap \overline{\Gamma}_D\neq\emptyset\). On \(\Gamma_D\) and \(\Gamma_N\) we place the homogeneous Dirichlet and Neumann boundary conditions, respectively. Due to lack of local regularity, we used the elliptic decomposition of the solution as did P. Grisvard [J. Math. Pures Appl. (9) 68, No. 2, 215–259 (1989; Zbl 0683.49012)] combined with the use of appropriated cutoff functions.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L05 Wave equation
35B35 Stability in context of PDEs
35L70 Second-order nonlinear hyperbolic equations

Citations:

Zbl 0683.49012
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), pp. 61–105. · Zbl 1107.35077
[2] 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 Equations, 248 (2010), pp. 1473–1517. · Zbl 1397.35068
[3] F. Alabau-Boussouira, New trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems, J. Differential Equations, 249 (2010), pp. 1145–1178. · Zbl 1201.35042
[4] F. Alabau-Boussouira and Kaïs Ammari, Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system, J. Funct. Anal., 260 (2011) 2424–2450. · Zbl 1217.93034
[5] F. Alabau-Boussouiraa, Y. Privat, and E. Trélatc, Nonlinear damped partial differential equations and their uniform discretizations, J. Funct. Anal., 273 (2017), pp. 352–403. · Zbl 1364.37155
[6] K. Ammari, S. Nicaise, and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay, Systems Control Lett., 59 (2010), pp. 623–628. · Zbl 1205.93126
[7] J. Banasiak and G. F. Roach On mixed boundary value problems of Dirichlet oblique-derivative type in plane domains with piecewise differentiable boundary, J. Differential Equations, 79 (1989), pp. 111–131. · Zbl 0698.35046
[8] V.Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International, Bucharest, 1976. · Zbl 0328.47035
[9] C. Bardos, G. Lebeau, and J. Rauch, Control and stabilisation de l’equation des ondes, in J. L. Lions controllabilité exacte des systèmes distribués, Rech. Math. Appl. 8, Masson, Paris, 1988, Appendix II. · Zbl 0644.49025
[10] R. Bey, J-P. Lohéac, and M. Moussaoui, Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilization of the wave equation, J. Math. Pures Appl. (9), 78 (1999), pp. 1043–1067. · Zbl 1054.35015
[11] N. Burq, Mesures semi-classiques et mesures de défaut, Séminaire Bourbaki, Presentation 826, Mathematical Society of France, Paris, 1997, pp. 167–195.
[12] N. Burq and P. Gérard, Contrôle Optimal des équations aux dérivées partielles, (2001).
[13] N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Math. Acad. Sci. Paris, 325 (1997), pp. 749–752. · Zbl 0906.93008
[14] N. Burq and J-M. Schlenker, Contrôle de equation des ondes dans des ouverts comportant des coins, Bull. Soc. Math. France, 126 (1998), pp. 601–637. · Zbl 0937.35097
[15] M. M. Cavalcanti, F. R. Dias Silva, and V. N. Domingos Cavalcanti, Uniform decay rates for the wave equation with nonlinear damping locally distributed in unbounded domains with finite measure, SIAM J. Control Optim., 52 (2014), pp. 545–580. · Zbl 1295.35080
[16] M. M. Cavalcanti, V. D. Domingos Cavalcanti, and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), pp. 407–459. · Zbl 1117.35048
[17] 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 (2009), pp. 4561–4580. · Zbl 1179.35052
[18] 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 (2010), pp. 925–964. · Zbl 1232.58019
[19] M. M. Cavalcanti, A. Khemmoudj, and M. Medjden, Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions, J. Math. Anal. Appl., 328 (2007), pp. 900–930. · Zbl 1107.35024
[20] M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, and D. Toundykov, Stabilization of the damped wave equation with Cauchy Ventcel boundary conditions, J. Evol. Equ., 9 (2009), pp. 143–169. · Zbl 1239.35020
[21] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pure Appl. (9), 58 (1979), pp. 249–274. · Zbl 0414.35044
[22] G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), pp. 106–113. · Zbl 0461.93036
[23] G. Chen, Control and stabilization for the wave equation in a bounded domain, Part II, SIAM J. Control Optim., 19 (1981), pp. 114–122. · Zbl 0461.93037
[24] M. Daoulatli, I. Lasiecka, and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), pp. 67–94. · Zbl 1172.35443
[25] P. Cornilleau, J. P. Lohéac, and A. Osses, Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers, J. Dyn. Control Syst., 16 (2010), pp. 163–188. · Zbl 1203.93172
[26] P. Gérard, Microlocal defect measures, Comm. Partial Differential Equations, 16 (1991) 1761–1794. · Zbl 0770.35001
[27] P. Grisvard, Controlabilité exacte des solutions de l’equation des ondes en présence de singularités, J. Math. Pures Appl. (9), 68 (1989), pp. 215–259. · Zbl 0683.49012
[28] P. Grisvard, Singularities in Boundary Value Problems, Springer, Berlin, 1992. · Zbl 0766.35001
[29] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. · Zbl 0695.35060
[30] V. Komornik, On the nonlinear boundary stabilization of the wave equation, Chin. Ann. Math. Ser. B, 14 (1993), pp. 153–164. · Zbl 0804.35065
[31] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. (9), 69 (1990), pp. 33–54. · Zbl 0636.93064
[32] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), pp. 163–182. · Zbl 0536.35043
[33] J. E. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), pp. 1250–1256. · Zbl 0657.93052
[34] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), pp. 507–533. · Zbl 0803.35088
[35] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal., 64 (2006), pp.1757–1797. · Zbl 1096.35021
[36] J. L. Lions, Controlabilité exacte, perturbations et stabilisation de systèmes distribuè, Vol. I, Masson, Paris, 1988.
[37] P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut., 12 (1999), pp. 251–283. · Zbl 0940.35034
[38] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), pp. 419–444. · Zbl 0923.35027
[39] L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation, SIAM J. Control Optim., 41 (2002), pp. 1554–1566. · Zbl 1032.35117
[40] M. Moussaoui, Singularies des solutions du probleme mele, controlabilite exacte et stabilisation frontiere ESAIM: Proceedings Elasticite, Viscoelasticite et Contrôle Optimal, 2 (1997), pp. 195–201. · Zbl 0892.35091
[41] M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation Math. Ann., 305 (1996), pp. 403–417. · Zbl 0856.35084
[42] M. Nakao, Decay of solutions of the wave equation with local degenerate dissipation, Israel J. Math., 95 (1996), pp. 25–42. · Zbl 0860.35072
[43] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), pp. 1561–1585. · Zbl 1180.35095
[44] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), pp. 935–958. · Zbl 1224.35247
[45] R. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, AMS, Providence, RI, 1997. · Zbl 0870.35004
[46] D. Toundykov, Optimal decay rates for solutions of nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponents source terms under mixed boundary, Nonlinear Anal., 67 (2007), pp. 512–544. · Zbl 1117.35050
[47] J. Vancostenoble and P. Martinez, Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks, SIAM J. Control Optim., 39 (2000), pp. 776–797. · Zbl 0984.35029
[48] W. L. Wendland, E. Stephan, G. C. Hsiao, and E. Meister, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Methods Appl. Sci., 1 (1979), pp. 265–321 · Zbl 0461.65082
[49] P.-F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim., 37 (1999), pp. 1568–1599. · Zbl 0951.35069
[50] P.-F. Yao, Modeling and Control in Vibrational and Structural Dynamics. A Differential Geometric Approach, Chapman Hall/CRC Appl. Math. Nonlinear Sci. Ser. CRC Press, Boca Raton, FL, 2011.
[51] P.-F. Yao, Observability inequalities for shallow shells, SIAM J. Control Optim., 38 (2000), pp. 1729–1756. · Zbl 0974.35013
[52] P.-F. Yao, Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Differential Equations, 241 (2007), pp 62–93. · Zbl 1214.35037
[53] P.-F. Yao, Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation, Chin. Ann. Math. Ser. B, 31 (2010), pp 59–70. · Zbl 1190.35036
[54] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), pp. 205–235. · Zbl 0716.35010
[55] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28 (1990), pp. 466–477. · Zbl 0695.93090
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.