# American Institute of Mathematical Sciences

June  2012, 32(6): 2315-2337. doi: 10.3934/dcds.2012.32.2315

## Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian

 1 Department of Mathematics & Statistics, Florida International University, Miami, FL 33199

Received  January 2011 Revised  May 2011 Published  February 2012

We consider an Euler-Bernoulli equation in a bounded domain with a local dissipation of viscoelastic type involving the $p$-Laplacian. The dissipation is effective in a suitable nonvoid subset of the domain under consideration. This equation corresponds to the plate equation with a localized structural damping when both the parameter $p$ and the space dimension equal two. First we prove existence, uniqueness, and smoothness results. Then, using an appropriate perturbed energy coupled with multiplier technique, we provide a constructive proof for the exponential and polynomial decay estimates of the underlying energy. It seems to us that this is the first time that a dissipation involving the $p$-Laplacian is used in the framework of stabilization of second order evolution equations with locally distributed damping.
Citation: Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315
##### References:
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Control and Opt., 30 (1992), 1024. Google Scholar [7] H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Mathématiques Appliquées pour la Maîtrise, (1983). Google Scholar [8] M. M. Cavalcanti, Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation,, Discrete Contin. Dyn. Syst., 8 (2002), 675. Google Scholar [9] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation,, Commun. Contemp. Math., 6 (2004), 705. Google Scholar [10] G. Chen, S. A. Fulling, F. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping,, SIAM J. Appl. Math., 51 (1991), 266. Google Scholar [11] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping,, Quart. Appl. Math., 39 (): 433. Google Scholar [12] S. Chen, K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping,, SIAM J. Appl. Math., 59 (1999), 651. Google Scholar [13] F. Conrad and B. Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback,, Asymptotic Anal., 7 (1993), 159. Google Scholar [14] C. M. Dafermos, Asymptotic behavior of solutions of evolution equations,, in, 40 (1978), 103. Google Scholar [15] B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Ann. Sci. École Norm. Sup., 36 (2003), 525. Google Scholar [16] A. Favini, M. A. Horn, I. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation,, Differential Integral Equations, 9 (1996), 267. Google Scholar [17] R. Benavides Guzmán and M. Tucsnak, Energy decay estimates for the damped plate equation with a local degenerated dissipation,, Systems & Control Letters, 48 (2003), 191. Google Scholar [18] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations,, J. Differential Equations, 59 (1985), 145. Google Scholar [19] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Port. Math., 46 (1989), 245. Google Scholar [20] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Recherches en Mathématiques Appliquées, 17 (1991). Google Scholar [21] A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems,, Arch. Rational Mech. Anal., 100 (1988), 191. Google Scholar [22] G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plate with dissipation acting only via moments-limiting behavior,, J.M.A.A., 229 (1999), 452. Google Scholar [23] V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method,", RAM: Research in Applied Mathematics, (1994). Google Scholar [24] V. Komornik, Decay estimates for the wave equation with internal damping,, in, 118 (1994), 253. Google Scholar [25] V. Komornik and S. Kouémou-Patcheu, Well-posedness and decay estimates for a Petrovsky system with internal damping,, Adv. Math. Sci. Appl., 7 (1997), 245. Google Scholar [26] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J.M.P.A., 69 (1990), 33. Google Scholar [27] S. Kouémou Patcheu, "Stabilisation Interne de Certains Systèmes Distribués,", Ph.D thesis, (1995). Google Scholar [28] J. Lagnese, Control of wave processes with distributed control supported on a subregion,, SIAM J. Control and Opt., 21 (1983), 68. Google Scholar [29] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Studies in Applied Mathematics, 10 (1989). Google Scholar [30] I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only,, J. Differential Equations, 95 (1992), 169. Google Scholar [31] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms-an intrinsic approach,, in, 252 (2007). Google Scholar [32] I. Lasiecka and R. Triggiani, Uniform energy decay rates of hyperbolic equations with nonlinear boundary and interior dissipation,, Control Cybernet, 37 (2008), 935. Google Scholar [33] G. Lebeau, Équation des ondes amorties,, in, 19 (1996), 73. Google Scholar [34] J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués,", Vol. 1, 8 (1988). Google Scholar [35] J.-L. Lions, "Quelques Méthodes de Résolutions des Problèmes aux Limites Non Linéaires,", Dunod, (1969). Google Scholar [36] K. Liu, Locally distributed control and damping for the conservative systems,, SIAM J. Control and Opt., 35 (1997), 1574. Google Scholar [37] K. Liu and Z. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,, SIAM J. Control and Opt., 36 (1998), 1086. Google Scholar [38] P. Martinez, "Stabilisation de Systèmes Distribués Semilinéaires: Domaines Presque Étoilés et Inégalités Intégrales Généralisées,", Ph.D thesis, (1998). Google Scholar [39] M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation,, Math. Ann., 305 (1996), 403. Google Scholar [40] L. Nirenberg, On elliptic partial differential equations,, Annali della Scuola Normale Superiore di Pisa (3), 13 (1959), 115. Google Scholar [41] J. Simon, Compact sets in the space $L^ p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. Google Scholar [42] M. Slemrod, Weak asymptotic decay via a "relaxed invariance principle" for a wave equation with nonlinear, nonmonotone damping,, Proc. Royal Soc. Edinburgh Sect. A, 113 (1989), 87. Google Scholar [43] L. R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement non linéaire localisé,, C. R. Acad. Paris Série I Math., 325 (1997), 1175. Google Scholar [44] L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping,, J.D.E., 145 (1998), 502. Google Scholar [45] L. R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with L$^r$ localizing coefficient,, Comm. in P.D.E., 23 (1998), 1839. Google Scholar [46] L. R. Tcheugoué Tébou, Energy decay estimates for the damped Euler-Bernoulli equation with an unbounded localizing coefficient,, Portugal. Math. (N.S.), 61 (2004), 375. Google Scholar [47] L. R. Tcheugoué Tébou, A direct method for the stabilization of some locally damped semilinear wave equations,, C. R. Math. Acad. Sci. Paris, 342 (2006), 859. Google Scholar [48] L. Tebou, A Carleman estimates based method for the stabilization of some locally damped semilinear hyperbolic equations,, ESAIM Control Optim. Calc. Var., 14 (2008), 561. Google Scholar [49] L. Tebou, Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping,, Nonlinear Anal., 71 (2009). Google Scholar [50] D. Toundykov, 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 (2007), 512. Google Scholar [51] M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation,, Math. Methods Appl. Sci., 19 (1996), 897. Google Scholar [52] H. Zhao, K. Liu and Z. Liu, A note on the exponential decay of energy of a Euler-Bernoulli beam with local viscoelasticity,, J. Elasticity, 74 (2004), 175. Google Scholar [53] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Commun. P.D.E., 15 (1990), 205. Google Scholar [54] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains,, J. Math. Pures. Appl., 70 (1991), 513. Google Scholar

show all references

##### References:
 [1] S. Agmon, The $L_p$ approach to the Dirichlet problem. I. Regularity theorems,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 405. Google Scholar [2] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems,, Appl. Math. Optim., 51 (2005), 61. Google Scholar [3] F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation,, J. Evol. Equ., 6 (2006), 95. Google Scholar [4] H. T. Banks, R. C. Smith and Y. Wang, The modeling of piezoceramic patch interactions with shells, plates, and beams,, Quart. Appl. Math., 53 (1995), 353. Google Scholar [5] V. Barbu, "Analysis and Control of Nonlinear Infinite-Dimensional Systems,", Mathematics in Science and Engineering, 190 (1993). Google Scholar [6] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control and Opt., 30 (1992), 1024. Google Scholar [7] H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Mathématiques Appliquées pour la Maîtrise, (1983). Google Scholar [8] M. M. Cavalcanti, Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation,, Discrete Contin. Dyn. Syst., 8 (2002), 675. Google Scholar [9] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation,, Commun. Contemp. Math., 6 (2004), 705. Google Scholar [10] G. Chen, S. A. Fulling, F. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping,, SIAM J. Appl. Math., 51 (1991), 266. Google Scholar [11] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping,, Quart. Appl. Math., 39 (): 433. Google Scholar [12] S. Chen, K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping,, SIAM J. Appl. Math., 59 (1999), 651. Google Scholar [13] F. Conrad and B. Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback,, Asymptotic Anal., 7 (1993), 159. Google Scholar [14] C. M. Dafermos, Asymptotic behavior of solutions of evolution equations,, in, 40 (1978), 103. Google Scholar [15] B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Ann. Sci. École Norm. Sup., 36 (2003), 525. Google Scholar [16] A. Favini, M. A. Horn, I. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation,, Differential Integral Equations, 9 (1996), 267. Google Scholar [17] R. Benavides Guzmán and M. Tucsnak, Energy decay estimates for the damped plate equation with a local degenerated dissipation,, Systems & Control Letters, 48 (2003), 191. Google Scholar [18] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations,, J. Differential Equations, 59 (1985), 145. Google Scholar [19] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Port. Math., 46 (1989), 245. Google Scholar [20] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Recherches en Mathématiques Appliquées, 17 (1991). Google Scholar [21] A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems,, Arch. Rational Mech. Anal., 100 (1988), 191. Google Scholar [22] G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plate with dissipation acting only via moments-limiting behavior,, J.M.A.A., 229 (1999), 452. Google Scholar [23] V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method,", RAM: Research in Applied Mathematics, (1994). Google Scholar [24] V. Komornik, Decay estimates for the wave equation with internal damping,, in, 118 (1994), 253. Google Scholar [25] V. Komornik and S. Kouémou-Patcheu, Well-posedness and decay estimates for a Petrovsky system with internal damping,, Adv. Math. Sci. Appl., 7 (1997), 245. Google Scholar [26] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J.M.P.A., 69 (1990), 33. Google Scholar [27] S. Kouémou Patcheu, "Stabilisation Interne de Certains Systèmes Distribués,", Ph.D thesis, (1995). Google Scholar [28] J. Lagnese, Control of wave processes with distributed control supported on a subregion,, SIAM J. Control and Opt., 21 (1983), 68. Google Scholar [29] J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Studies in Applied Mathematics, 10 (1989). Google Scholar [30] I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only,, J. Differential Equations, 95 (1992), 169. Google Scholar [31] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms-an intrinsic approach,, in, 252 (2007). Google Scholar [32] I. Lasiecka and R. Triggiani, Uniform energy decay rates of hyperbolic equations with nonlinear boundary and interior dissipation,, Control Cybernet, 37 (2008), 935. Google Scholar [33] G. Lebeau, Équation des ondes amorties,, in, 19 (1996), 73. Google Scholar [34] J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués,", Vol. 1, 8 (1988). Google Scholar [35] J.-L. Lions, "Quelques Méthodes de Résolutions des Problèmes aux Limites Non Linéaires,", Dunod, (1969). Google Scholar [36] K. Liu, Locally distributed control and damping for the conservative systems,, SIAM J. Control and Opt., 35 (1997), 1574. Google Scholar [37] K. Liu and Z. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,, SIAM J. Control and Opt., 36 (1998), 1086. Google Scholar [38] P. Martinez, "Stabilisation de Systèmes Distribués Semilinéaires: Domaines Presque Étoilés et Inégalités Intégrales Généralisées,", Ph.D thesis, (1998). Google Scholar [39] M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation,, Math. Ann., 305 (1996), 403. Google Scholar [40] L. Nirenberg, On elliptic partial differential equations,, Annali della Scuola Normale Superiore di Pisa (3), 13 (1959), 115. Google Scholar [41] J. Simon, Compact sets in the space $L^ p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. Google Scholar [42] M. Slemrod, Weak asymptotic decay via a "relaxed invariance principle" for a wave equation with nonlinear, nonmonotone damping,, Proc. Royal Soc. Edinburgh Sect. A, 113 (1989), 87. Google Scholar [43] L. R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement non linéaire localisé,, C. R. Acad. Paris Série I Math., 325 (1997), 1175. Google Scholar [44] L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping,, J.D.E., 145 (1998), 502. Google Scholar [45] L. R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with L$^r$ localizing coefficient,, Comm. in P.D.E., 23 (1998), 1839. Google Scholar [46] L. R. Tcheugoué Tébou, Energy decay estimates for the damped Euler-Bernoulli equation with an unbounded localizing coefficient,, Portugal. Math. (N.S.), 61 (2004), 375. Google Scholar [47] L. R. Tcheugoué Tébou, A direct method for the stabilization of some locally damped semilinear wave equations,, C. R. Math. Acad. Sci. Paris, 342 (2006), 859. Google Scholar [48] L. Tebou, A Carleman estimates based method for the stabilization of some locally damped semilinear hyperbolic equations,, ESAIM Control Optim. Calc. Var., 14 (2008), 561. Google Scholar [49] L. Tebou, Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping,, Nonlinear Anal., 71 (2009). Google Scholar [50] D. Toundykov, 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 (2007), 512. Google Scholar [51] M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation,, Math. Methods Appl. Sci., 19 (1996), 897. Google Scholar [52] H. Zhao, K. Liu and Z. Liu, A note on the exponential decay of energy of a Euler-Bernoulli beam with local viscoelasticity,, J. Elasticity, 74 (2004), 175. Google Scholar [53] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Commun. P.D.E., 15 (1990), 205. Google Scholar [54] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains,, J. Math. Pures. Appl., 70 (1991), 513. Google Scholar
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