American Institute of Mathematical Sciences

Advanced
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

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

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

[1]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021
[2]

Valentin Keyantuo, Louis Tebou, Mahamadi Warma. A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2875-2889. doi: 10.3934/dcds.2020152
[3]

Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425
[4]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029
[5]

Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675
[6]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45
[7]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723
[8]

Kaïs Ammari, Denis Mercier, Virginie Régnier, Julie Valein. Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Communications on Pure & Applied Analysis, 2012, 11 (2) : 785-807. doi: 10.3934/cpaa.2012.11.785
[9]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[10]

Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks & Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709
[11]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251
[12]

Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303
[13]

Kerstin Does. An evolution equation involving the normalized $P$-Laplacian. Communications on Pure & Applied Analysis, 2011, 10 (1) : 361-396. doi: 10.3934/cpaa.2011.10.361
[14]

Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110
[15]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
[16]

Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029
[17]

Jacques Giacomoni, Tuhina Mukherjee, Konijeti Sreenadh. Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 311-337. doi: 10.3934/dcdss.2019022
[18]

Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016
[19]

Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033
[20]

Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences