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

Louis Tebou

doi:10.3934/dcds.2012.32.2315

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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

Louis Tebou ^1,

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

Received January 2011 Revised May 2011 Published February 2012

Abstract
Related Papers

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.

Keywords: differential inequalities, plate equation, Euler-Bernoulli equation, localized damping, Lyapunov method., stabilization, $p$-Laplacian, multiplier techniques.

Mathematics Subject Classification: Primary: 93D15; Secondary: 35D30, 35D35, 35Q74, 37L15, 74K2.

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

References:

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[2]	F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105. Google Scholar
[3]	F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation, J. Evol. Equ., 6 (2006), 95-112. 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), 353-381. Google Scholar
[5]	V. Barbu, "Analysis and Control of Nonlinear Infinite-Dimensional Systems," Mathematics in Science and Engineering, 190, Academic Press, Inc., Boston, MA, 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), 1024-1065. Google Scholar
[7]	H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 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), 675-695. 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), 705-731. 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), 266-301. 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), 651-668. 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), 159-177. Google Scholar
[14]	C. M. Dafermos, Asymptotic behavior of solutions of evolution equations, in "Nonlinear Evolution Equations" (ed. M.G. Crandall) (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), Publ. Math. Res. Center Univ. Wisconsin, 40, Academic Press, New York-London, (1978), 103-123. 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), 525-551. 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), 267-294. 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), 191-197. Google Scholar
[18]	A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59 (1985), 145-154. Google Scholar
[19]	A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math., 46 (1989), 245-258. Google Scholar
[20]	A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliquées, 17, Masson, Paris, 1991. Google Scholar
[21]	A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal., 100 (1988), 191-206. 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), 452-479. Google Scholar
[23]	V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method," RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994. Google Scholar
[24]	V. Komornik, Decay estimates for the wave equation with internal damping, in "Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena" (Vorau, 1993), International Series of Num. Math., 118, Birkhäuser, Basel, (1994), 253-266. 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), 245-260. 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), 33-54. Google Scholar
[27]	S. Kouémou Patcheu, "Stabilisation Interne de Certains Systèmes Distribués," Ph.D thesis, Université de Strasbourg I (Louis Pasteur), Strasbourg, 1995; Prépublication de l'Institut de Recherche Mathématique Avancée, 1995/19, Université Louis Pasteur, Département de Mathématique, Institut de Recherche Mathématique Avancée, Strasbourg, 1995. Google Scholar
[28]	J. Lagnese, Control of wave processes with distributed control supported on a subregion, SIAM J. Control and Opt., 21 (1983), 68-85. Google Scholar
[29]	J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 169-182. 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 "Free and Moving Boundaries," Lect. Notes Pure Appl. Math., 252, Chapman & Hall/CRC, Boca Raton, FL, 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), 935-969. Google Scholar
[33]	G. Lebeau, Équation des ondes amorties, in "Algebraic and Geometric Methods in Mathematical Physics" (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, (1996), 73-109. Google Scholar
[34]	J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués," Vol. 1, Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8, Masson, Paris, 1988. Google Scholar
[35]	J.-L. Lions, "Quelques Méthodes de Résolutions des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. Google Scholar
[36]	K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control and Opt., 35 (1997), 1574-1590. 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), 1086-1098. Google Scholar
[38]	P. Martinez, "Stabilisation de Systèmes Distribués Semilinéaires: Domaines Presque Étoilés et Inégalités Intégrales Généralisées," Ph.D thesis, Université Louis Pasteur (Strasbourg I), Strasbourg, 1998; Prépublication de l'Institut de Recherche Mathématique Avancée, 1998/44, Université Louis Pasteur, Département de Mathématique, Institut de Recherche Mathématique Avancée, Strasbourg, 1998. Google Scholar
[39]	M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417. Google Scholar
[40]	L. Nirenberg, On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa (3), 13 (1959), 115-162. Google Scholar
[41]	J. Simon, Compact sets in the space $L^ p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. 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), 87-97. 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), 1175-1179. Google Scholar
[44]	L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping, J.D.E., 145 (1998), 502-524. 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), 1839-1855. 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), 375-391. 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), 859-864. 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), 561-574. Google Scholar
[49]	L. Tebou, Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping, Nonlinear Anal., 71 (2009), e2288-e2297. 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), 512-544. Google Scholar
[51]	M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation, Math. Methods Appl. Sci., 19 (1996), 897-907. 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), 175-183. Google Scholar
[53]	E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Commun. P.D.E., 15 (1990), 205-235. Google Scholar
[54]	E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures. Appl., 70 (1991), 513-529. 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), 405-448. 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), 61-105. Google Scholar
[3]	F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation, J. Evol. Equ., 6 (2006), 95-112. 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), 353-381. Google Scholar
[5]	V. Barbu, "Analysis and Control of Nonlinear Infinite-Dimensional Systems," Mathematics in Science and Engineering, 190, Academic Press, Inc., Boston, MA, 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), 1024-1065. Google Scholar
[7]	H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 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), 675-695. 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), 705-731. 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), 266-301. 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), 651-668. 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), 159-177. Google Scholar
[14]	C. M. Dafermos, Asymptotic behavior of solutions of evolution equations, in "Nonlinear Evolution Equations" (ed. M.G. Crandall) (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), Publ. Math. Res. Center Univ. Wisconsin, 40, Academic Press, New York-London, (1978), 103-123. 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), 525-551. 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), 267-294. 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), 191-197. Google Scholar
[18]	A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59 (1985), 145-154. Google Scholar
[19]	A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math., 46 (1989), 245-258. Google Scholar
[20]	A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliquées, 17, Masson, Paris, 1991. Google Scholar
[21]	A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal., 100 (1988), 191-206. 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), 452-479. Google Scholar
[23]	V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method," RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994. Google Scholar
[24]	V. Komornik, Decay estimates for the wave equation with internal damping, in "Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena" (Vorau, 1993), International Series of Num. Math., 118, Birkhäuser, Basel, (1994), 253-266. 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), 245-260. 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), 33-54. Google Scholar
[27]	S. Kouémou Patcheu, "Stabilisation Interne de Certains Systèmes Distribués," Ph.D thesis, Université de Strasbourg I (Louis Pasteur), Strasbourg, 1995; Prépublication de l'Institut de Recherche Mathématique Avancée, 1995/19, Université Louis Pasteur, Département de Mathématique, Institut de Recherche Mathématique Avancée, Strasbourg, 1995. Google Scholar
[28]	J. Lagnese, Control of wave processes with distributed control supported on a subregion, SIAM J. Control and Opt., 21 (1983), 68-85. Google Scholar
[29]	J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 169-182. 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 "Free and Moving Boundaries," Lect. Notes Pure Appl. Math., 252, Chapman & Hall/CRC, Boca Raton, FL, 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), 935-969. Google Scholar
[33]	G. Lebeau, Équation des ondes amorties, in "Algebraic and Geometric Methods in Mathematical Physics" (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, (1996), 73-109. Google Scholar
[34]	J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués," Vol. 1, Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8, Masson, Paris, 1988. Google Scholar
[35]	J.-L. Lions, "Quelques Méthodes de Résolutions des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. Google Scholar
[36]	K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control and Opt., 35 (1997), 1574-1590. 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), 1086-1098. Google Scholar
[38]	P. Martinez, "Stabilisation de Systèmes Distribués Semilinéaires: Domaines Presque Étoilés et Inégalités Intégrales Généralisées," Ph.D thesis, Université Louis Pasteur (Strasbourg I), Strasbourg, 1998; Prépublication de l'Institut de Recherche Mathématique Avancée, 1998/44, Université Louis Pasteur, Département de Mathématique, Institut de Recherche Mathématique Avancée, Strasbourg, 1998. Google Scholar
[39]	M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417. Google Scholar
[40]	L. Nirenberg, On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa (3), 13 (1959), 115-162. Google Scholar
[41]	J. Simon, Compact sets in the space $L^ p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. 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), 87-97. 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), 1175-1179. Google Scholar
[44]	L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping, J.D.E., 145 (1998), 502-524. 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), 1839-1855. 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), 375-391. 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), 859-864. 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), 561-574. Google Scholar
[49]	L. Tebou, Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping, Nonlinear Anal., 71 (2009), e2288-e2297. 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), 512-544. Google Scholar
[51]	M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation, Math. Methods Appl. Sci., 19 (1996), 897-907. 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), 175-183. Google Scholar
[53]	E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Commun. P.D.E., 15 (1990), 205-235. Google Scholar
[54]	E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures. Appl., 70 (1991), 513-529. Google Scholar

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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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