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Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*
1. | Department of Mathematics, State University of Londrina, Londrina, 86057-970, Brazil |
2. | Center of Exact Sciences, State University of Mato Grosso do Sul, Dourados, 79804-970, Brazil |
In this paper we consider new results on well-posedness and long-time dynamics for a class of extensible beam/plate models whose dissipative effect is given by the product of two nonlinear terms. The addressed model contains a nonlocal nonlinear damping term which generalizes some classes of dissipations usually given in the literature, namely, the linear, the nonlinear and the nonlocal frictional ones. A first mathematical analysis of such damping term is presented and represents the main novelty in our approach.
References:
[1] |
A. V. Balakrishnan and L. W. Taylor,
Distributed Parameter Nonlinear Damping Models for Flight Structures in: Proceedings ''Daming 89", Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. |
[2] |
A. C. Biazutti and H. R. Crippa,
Global attractor and inertial set for the beam equation, Appl. Anal., 55 (1994), 61-78.
doi: 10.1080/00036819408840290. |
[3] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and T. F. Ma,
Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differential Integral Equations, 17 (2004), 495-510.
|
[4] |
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.
doi: 10.1142/S0219199704001483. |
[5] |
R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto,
Asymptotic behavior of a Bernoulli-Euler type equation with nonlinear localized damping, Contributions to Nonlinear Analysis -Progress in nonlinear partial differential equations and their applications, 66 (2005), 67-91.
doi: 10.1007/3-7643-7401-2_5. |
[6] |
I. Chueshov,
Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.
|
[7] |
I. Chueshov,
Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.
doi: 10.1016/j.jde.2011.08.022. |
[8] |
I. Chueshov and S. Kolbasin,
Plate models with state-dependent damping coefficient and their quasi-static limits, Nonlinear Anal., 73 (2010), 1626-1644.
doi: 10.1016/j.na.2010.04.072. |
[9] |
I. Chueshov and S. Kolbasin,
Long-time dynamics in plate models with strong nonlinear damping, Commun. Pure Appl. Anal., 11 (2012), 659-674.
doi: 10.3934/cpaa.2012.11.659. |
[10] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping Mem. Amer. Math. Soc., 195 (2008), viii+183 pp.
doi: 10.1090/memo/0912. |
[11] |
I. Chueshov and I. Lasiecka,
Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[12] |
I. Chueshov, I. Lasiecka and D. Toundykov,
Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509.
doi: 10.3934/dcds.2008.20.459. |
[13] |
M. Coti Zelati,
Global and exponential attractors for the singularly perturbed extensible beam, Discrete Contin. Dyn. Syst., 25 (2009), 1041-1060.
doi: 10.3934/dcds.2009.25.1041. |
[14] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.
doi: 10.1090/surv/025. |
[15] |
M. A. Jorge Silva and V. Narciso,
Long-time behavior for a plate equation with nonlocal weak damping, Differential Integral Equations, 27 (2014), 931-948.
|
[16] |
M. A. Jorge Silva and V. Narciso,
Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst., 35 (2015), 985-1008.
doi: 10.3934/dcds.2015.35.985. |
[17] |
J. R. Kang,
Global attractor for an extensible beam equation with localized nonlinear damping and linear memory, Math. Methods Appl. Sci., 34 (2011), 1430-1439.
doi: 10.1002/mma.1450. |
[18] |
A. Kh. Khanmamedov,
Global attractors for the plate equation with a localized damping and critical exponent in an unbounded domain, J. Differential Equation, 225 (2006), 528-548.
doi: 10.1016/j.jde.2005.12.001. |
[19] |
A. Kh. Khanmamedov,
Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[20] |
A. Kh. Khanmamedov,
A global attractor for the plate equation with displacement-dependent damping, Nonlinear Anal., 74 (2011), 1607-1615.
doi: 10.1016/j.na.2010.10.031. |
[21] |
S. Kolbasin,
Attractors for Kirchhoff's equation with a nonlinear damping coefficient, Nonlinear Anal., 71 (2009), 2361-2371.
doi: 10.1016/j.na.2009.01.187. |
[22] |
S. Kouémou Patcheu,
On a global solution and asymptotic behaviour for the generalized damped extensible beam equation, J. Differential Equations, 135 (1997), 299-314.
doi: 10.1006/jdeq.1996.3231. |
[23] |
H. Lange and G. Perla Menzala,
Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092.
|
[24] |
P. Lazo,
Global solutions for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601.
doi: 10.1016/j.amc.2007.11.056. |
[25] |
J. -L. Lions,
Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires Dunod, Paris, 1969. |
[26] |
J. -L. Lions and E. Magenes,
Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972. |
[27] |
T. F. Ma and V. Narciso,
Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Anal., 73 (2010), 3402-3412.
doi: 10.1016/j.na.2010.07.023. |
[28] |
T. F. Ma, V. Narciso and M. L. Pelicer,
Long-time behavior of a model of extensible beams with nonlinear boundary dissipations, J. Math. Anal. Appl., 396 (2012), 694-703.
doi: 10.1016/j.jmaa.2012.07.004. |
[29] |
L. A. Medeiros and M. Milla Miranda,
On a nonlinear wave equation with damping, Rev. Mat. Univ. Complut. Madrid, 3 (1990), 213-231.
|
[30] |
M. Nakao,
Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (1976), 257-265.
doi: 10.2206/kyushumfs.30.257. |
[31] |
M. Nakao,
On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions, Math. Z., 193 (1986), 227-234.
doi: 10.1007/BF01174332. |
[32] |
M. Nakao,
Global attractors for wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229.
doi: 10.1016/j.jde.2005.09.013. |
[33] |
M. Potomkin,
Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192.
doi: 10.3934/cpaa.2010.9.161. |
[34] |
J. Simon,
Compact sets in the space $$L^{p}(0,T;B)$ $, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[35] |
C. F. Vasconcellos and L. M. Teixeira,
Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping, Ann. Fac. Sci. Toulouse Math., 8 (1999), 173-193.
|
[36] |
D. Wang and J. Zhang,
Global attractor for a nonlinear plate equation with supported boundary conditions, J. Math. Anal. Appl., 363 (2010), 468-480.
doi: 10.1016/j.jmaa.2009.09.020. |
[37] |
Y. Zhijian,
On an extensible beam equation with nonlinear damping and source terms, J. Differential Equations, 254 (2013), 3903-3927.
doi: 10.1016/j.jde.2013.02.008. |
show all references
References:
[1] |
A. V. Balakrishnan and L. W. Taylor,
Distributed Parameter Nonlinear Damping Models for Flight Structures in: Proceedings ''Daming 89", Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. |
[2] |
A. C. Biazutti and H. R. Crippa,
Global attractor and inertial set for the beam equation, Appl. Anal., 55 (1994), 61-78.
doi: 10.1080/00036819408840290. |
[3] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and T. F. Ma,
Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differential Integral Equations, 17 (2004), 495-510.
|
[4] |
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.
doi: 10.1142/S0219199704001483. |
[5] |
R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto,
Asymptotic behavior of a Bernoulli-Euler type equation with nonlinear localized damping, Contributions to Nonlinear Analysis -Progress in nonlinear partial differential equations and their applications, 66 (2005), 67-91.
doi: 10.1007/3-7643-7401-2_5. |
[6] |
I. Chueshov,
Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.
|
[7] |
I. Chueshov,
Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.
doi: 10.1016/j.jde.2011.08.022. |
[8] |
I. Chueshov and S. Kolbasin,
Plate models with state-dependent damping coefficient and their quasi-static limits, Nonlinear Anal., 73 (2010), 1626-1644.
doi: 10.1016/j.na.2010.04.072. |
[9] |
I. Chueshov and S. Kolbasin,
Long-time dynamics in plate models with strong nonlinear damping, Commun. Pure Appl. Anal., 11 (2012), 659-674.
doi: 10.3934/cpaa.2012.11.659. |
[10] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping Mem. Amer. Math. Soc., 195 (2008), viii+183 pp.
doi: 10.1090/memo/0912. |
[11] |
I. Chueshov and I. Lasiecka,
Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[12] |
I. Chueshov, I. Lasiecka and D. Toundykov,
Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509.
doi: 10.3934/dcds.2008.20.459. |
[13] |
M. Coti Zelati,
Global and exponential attractors for the singularly perturbed extensible beam, Discrete Contin. Dyn. Syst., 25 (2009), 1041-1060.
doi: 10.3934/dcds.2009.25.1041. |
[14] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.
doi: 10.1090/surv/025. |
[15] |
M. A. Jorge Silva and V. Narciso,
Long-time behavior for a plate equation with nonlocal weak damping, Differential Integral Equations, 27 (2014), 931-948.
|
[16] |
M. A. Jorge Silva and V. Narciso,
Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst., 35 (2015), 985-1008.
doi: 10.3934/dcds.2015.35.985. |
[17] |
J. R. Kang,
Global attractor for an extensible beam equation with localized nonlinear damping and linear memory, Math. Methods Appl. Sci., 34 (2011), 1430-1439.
doi: 10.1002/mma.1450. |
[18] |
A. Kh. Khanmamedov,
Global attractors for the plate equation with a localized damping and critical exponent in an unbounded domain, J. Differential Equation, 225 (2006), 528-548.
doi: 10.1016/j.jde.2005.12.001. |
[19] |
A. Kh. Khanmamedov,
Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[20] |
A. Kh. Khanmamedov,
A global attractor for the plate equation with displacement-dependent damping, Nonlinear Anal., 74 (2011), 1607-1615.
doi: 10.1016/j.na.2010.10.031. |
[21] |
S. Kolbasin,
Attractors for Kirchhoff's equation with a nonlinear damping coefficient, Nonlinear Anal., 71 (2009), 2361-2371.
doi: 10.1016/j.na.2009.01.187. |
[22] |
S. Kouémou Patcheu,
On a global solution and asymptotic behaviour for the generalized damped extensible beam equation, J. Differential Equations, 135 (1997), 299-314.
doi: 10.1006/jdeq.1996.3231. |
[23] |
H. Lange and G. Perla Menzala,
Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092.
|
[24] |
P. Lazo,
Global solutions for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601.
doi: 10.1016/j.amc.2007.11.056. |
[25] |
J. -L. Lions,
Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires Dunod, Paris, 1969. |
[26] |
J. -L. Lions and E. Magenes,
Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972. |
[27] |
T. F. Ma and V. Narciso,
Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Anal., 73 (2010), 3402-3412.
doi: 10.1016/j.na.2010.07.023. |
[28] |
T. F. Ma, V. Narciso and M. L. Pelicer,
Long-time behavior of a model of extensible beams with nonlinear boundary dissipations, J. Math. Anal. Appl., 396 (2012), 694-703.
doi: 10.1016/j.jmaa.2012.07.004. |
[29] |
L. A. Medeiros and M. Milla Miranda,
On a nonlinear wave equation with damping, Rev. Mat. Univ. Complut. Madrid, 3 (1990), 213-231.
|
[30] |
M. Nakao,
Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (1976), 257-265.
doi: 10.2206/kyushumfs.30.257. |
[31] |
M. Nakao,
On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions, Math. Z., 193 (1986), 227-234.
doi: 10.1007/BF01174332. |
[32] |
M. Nakao,
Global attractors for wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229.
doi: 10.1016/j.jde.2005.09.013. |
[33] |
M. Potomkin,
Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192.
doi: 10.3934/cpaa.2010.9.161. |
[34] |
J. Simon,
Compact sets in the space $$L^{p}(0,T;B)$ $, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[35] |
C. F. Vasconcellos and L. M. Teixeira,
Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping, Ann. Fac. Sci. Toulouse Math., 8 (1999), 173-193.
|
[36] |
D. Wang and J. Zhang,
Global attractor for a nonlinear plate equation with supported boundary conditions, J. Math. Anal. Appl., 363 (2010), 468-480.
doi: 10.1016/j.jmaa.2009.09.020. |
[37] |
Y. Zhijian,
On an extensible beam equation with nonlinear damping and source terms, J. Differential Equations, 254 (2013), 3903-3927.
doi: 10.1016/j.jde.2013.02.008. |
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