Uniform stabilization for the semi-linear wave equation with nonlinear Kelvin-Voigt damping. (English) Zbl 07930228
Summary: This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin-Voigt type which is distributed around a neighborhood of the boundary and the second is a frictional damping depending in the first one. We show uniform decay rate results of the corresponding energy for all initial data taken in bounded sets of finite energy phase-space. The proof is based on obtaining an observability inequality which combines unique continuation properties and the tools of the Microlocal Analysis Theory.
MSC:
| 35-XX | Partial differential equations |
Keywords:
uniform stabilization; semilinear wave equation; nonlinear Kelvin-Voigt damping; viscoelastic feedbackReferences:
| [1] | Alabau-Boussouira, F., 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. Differ. Equ., 248, 1473-1517, 2010 · Zbl 1397.35068 · doi:10.1016/j.jde.2009.12.005 |
| [2] | Ammari, K.; Hassine, F.; Robbiano, L., Stabilization for the wave equation with singular Kelvin-Voigt damping, Arch. Ration. Mech. Anal., 236, 577-601, 2020 · Zbl 1436.93113 · doi:10.1007/s00205-019-01476-4 |
| [3] | Ammari, K.; Hassine, F., Stabilization of Kelvin-Voigt Damped Systems, Advances in Mechanics and Mathematics, 2022, Cham: Birkhäuser, Cham · Zbl 1512.35001 |
| [4] | Ammari, K.; Liu, Z.; Shel, F., Stability of the wave equations on a tree with local Kelvin-Voigt damping, Semigroup Forum, 100, 2, 364-382, 2020 · Zbl 1433.35388 · doi:10.1007/s00233-019-10064-7 |
| [5] | Barbu, V., Semigroup Approach to Nonlinear Diffusion Equations, 2021, Singapore: World Scientific, Singapore · Zbl 1528.35001 · doi:10.1142/12534 |
| [6] | Bardos, C.; Lebeau, G.; Rauch, J., Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30, 1024-1065, 1992 · Zbl 0786.93009 · doi:10.1137/0330055 |
| [7] | Burq, N.; Gérard, P., Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math., 325, 749-752, 1997 · Zbl 0906.93008 · doi:10.1016/S0764-4442(97)80053-5 |
| [8] | Burq, N., Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14, 157-91, 1997 · Zbl 0892.93009 |
| [9] | Burq, N., Gérard, P.: Contrôle Optimal des équations aux dérivées partielles, http://www.math.u-psud.fr/ burq/articles/coursX.pdf (2001) |
| [10] | Cavalcanti, MM; Domingos Cavalcanti, VN; Fukuoka, R.; Soriano, JA, Uniform stabilization of the wave equation on compact surfaces and locally distributed damping methods, Appl. Anal., 15, 405-426, 2008 · Zbl 1183.35200 |
| [11] | Cavalcanti, MM; Domingos Cavalcanti, VN; Fukuoka, R.; Soriano, JA, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result, Arch. Ration. Mech. Anal., 197, 925-964, 2010 · Zbl 1232.58019 · doi:10.1007/s00205-009-0284-z |
| [12] | Cavalcanti, MM; Domingos Cavalcanti, VN; Fukuoka, R.; Soriano, JA, Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result, Trans. Am. Math. Soc., 361, 4561-4580, 2009 · Zbl 1179.35052 · doi:10.1090/S0002-9947-09-04763-1 |
| [13] | Cavalcanti, MM; Domingos Cavalcanti, VN; Fukuoka, R.; Pampu, AB; Astudillo, M., Uniform decay rate estimates for the semilinear wave equation in inhomogeneous medium with locally distributed nonlinear damping, Nonlinearity, 31, 4031-4064, 2018 · Zbl 1397.35025 · doi:10.1088/1361-6544/aac75d |
| [14] | Cavalcanti, MM; Domingos Cavalcanti, VN; Tebou, L., Stabilization of the wave equation with localized compensating frictional and Kelvin-Voigt dissipating mechanisms, Electron. J. Differ. Equ., 83, 18, 2017 · Zbl 1369.93479 |
| [15] | Cavalcanti, MM; Gonzalez Martinez, VH, Exponential decay for the semilinear wave equation with localized frictional and Kelvin-Voigt dissipating mechanisms, Asymptot. Anal., 128, 273-293, 2022 · Zbl 1506.35158 |
| [16] | Chueshov, I.; Eller, M.; Lasiecka, I., On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Commun. PDE, 27, 1901-51, 2002 · Zbl 1021.35020 · doi:10.1081/PDE-120016132 |
| [17] | Dehman, B.; Gérard, P.; Lebeau, G., Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254, 729-749, 2006 · Zbl 1127.93015 · doi:10.1007/s00209-006-0005-3 |
| [18] | Duyckaerts, T.; Zhang, X.; Zuazua, E., On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 1-41, 2008 · Zbl 1248.93031 · doi:10.1016/j.anihpc.2006.07.005 |
| [19] | Gérard, P., Microlocal defect measures, Commun. Partial Differ. Equ., 16, 1761-1794, 1991 · Zbl 0770.35001 · doi:10.1080/03605309108820822 |
| [20] | Haraux, A., Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differ. Equ., 59, 145-154, 1985 · Zbl 0535.35006 · doi:10.1016/0022-0396(85)90151-2 |
| [21] | Joly, R.; Laurent, C., Stabilization for the semilinear wave equation with geometric control, Anal. PDE, 6, 1089-1119, 2013 · Zbl 1329.35062 · doi:10.2140/apde.2013.6.1089 |
| [22] | Koch, H.; Tataru, D., Dispersive estimates for principally normal pseudodifferential operators, Commun. Pure Appl. Math., 58, 217-284, 2005 · Zbl 1078.35143 · doi:10.1002/cpa.20067 |
| [23] | Lasiecka, I.; Tataru, D., Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differ. Integr. Equ., 6, 507-533, 1993 · Zbl 0803.35088 |
| [24] | Liu, K.; Liu, Z., Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim., 36, 1086-1098, 1998 · Zbl 0909.35018 · doi:10.1137/S0363012996310703 |
| [25] | Liu, K.; Liu, Z., Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53, 265-280, 2002 · Zbl 0999.35012 · doi:10.1007/s00033-002-8155-6 |
| [26] | Liu, Z.; Rao, B., Exponential stability for wave equations with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 57, 419-432, 2006 · Zbl 1114.35023 · doi:10.1007/s00033-005-0029-2 |
| [27] | Martinez, P., A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut., 12, 251-283, 1999 · Zbl 0940.35034 · doi:10.5209/rev_REMA.1999.v12.n1.17227 |
| [28] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 1983, New York: Springer-Verlag, New York · Zbl 0516.47023 |
| [29] | Ruiz, A., Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl., 71, 455-467, 1992 · Zbl 0832.35084 |
| [30] | Simon, J., Compact sets in the space \(L^p(0; T;B)\), Ann. Mat. Pura Appl., 146, 65-96, 1987 · Zbl 0629.46031 · doi:10.1007/BF01762360 |
| [31] | Toundykov, D., Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions, Nonlinear Anal., 67, 2, 512-544, 2007 · Zbl 1117.35050 · doi:10.1016/j.na.2006.06.007 |
| [32] | Tebou, L.: A constructive method for the stabilization of the wave equation with localized Kelvi-Voigt damping. C. R. Acad. Sci. Paris Ser. I 350, 603-608 (2012) · Zbl 1255.35039 |
| [33] | Tebou, L., Sharp observability estimates for a system of two coupled nonconservative hyperbolic equations, Appl. Math. Optim., 66, 175-207, 2012 · Zbl 1269.93016 · doi:10.1007/s00245-012-9168-y |
| [34] | Zuazua, E., Exponential decay for semilinear wave equations with localized damping, Commun. Partial Differ. Equ., 15, 205-235, 1990 · Zbl 0716.35010 · doi:10.1080/03605309908820684 |
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.
