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July  2019, 24(7): 3281-3298. doi: 10.3934/dcdsb.2018320

On a beam model related to flight structures with nonlocal energy damping

Marcio A. Jorge Silva 1,, , Vando Narciso 2,†, and  André Vicente 3,‡,

1. 

Department of Mathematics, State University of Londrina, 86057-970, Londrina, PR, Brazil
2. 

Nucleus of Exact and Technological Sciences, State University of Mato Grosso do Sul, 79804-970, Dourados, MS, Brazil
3. 

Center of Exact and Technological Sciences, State University of Paraná West, 85819-110, Cascavel, PR, Brazil

* Corresponding author. M. A. Jorge Silva has been supported by CNPq, grant 441414/2014-1

† V. Narciso has been supported by FUNDECT, grant 219/2016

Received  November 2017 Revised  August 2018 Published  July 2019 Early access  January 2019

This paper deals with new results on existence, uniqueness and stability for a class of nonlinear beams arising in connection with nonlocal dissipative models for flight structures with energy damping first proposed by Balakrishnan-Taylor [2]. More precisely, the following
$ n $
-dimensional model is addressed
$ u_{tt}-\kappa \Delta u+\Delta ^2u-\gamma\left[\int_{\Omega}\left(|\Delta u|^2+|u_t|^2\right)dx \right]^q\Delta u_t+f(u) = 0 \ in \ \Omega \times \mathbb{R}^+, $
where
$ \Omega\subset \mathbb{R}^n $
 is a bounded domain with smooth boundary, the coefficient of extensibility
$ \kappa $
 is nonnegative, the damping coefficient
$ \gamma $
 is positive and
$ q\ge 1 $
. The nonlinear source
$ f(u) $
 can be seen as an external forcing term of lower order. Our main results feature global existence and uniqueness, polynomial stability and a non-exponential decay prospect.
Keywords: Extensible beams, flight structures, solution, stability.
Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 35L76.
Citation: Marcio A. Jorge Silva, Vando Narciso, André Vicente. On a beam model related to flight structures with nonlocal energy damping. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3281-3298. doi: 10.3934/dcdsb.2018320
References:
[1]

A. V. Balakrishnan, A theory of nonlinear damping in flexible structures, Stabilization of flexible structures, (1988), 1–12.

[2]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.

[3]

J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.

[4]

J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418.  doi: 10.1016/0022-0396(73)90056-9.

[5]

R. W. Bass and D. Zes, Spillover, Nonlinearity, and flexible structures, The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, NASA Conference Publication 10065 ed. L.W.Taylor, (1991), 1–14.

[6]

A. C. Biazutti and H. R. Crippa, Global attractor and inertial set for the beam equation, Applicable Analysis, 55 (1994), 61-78.  doi: 10.1080/00036819408840290.

[7]

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.

[8]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math., 195 (2008), ⅷ+183 pp. doi: 10.1090/memo/0912.

[9]

H. R. Clark, M. A. Rincon and R. D. Rodrigues, Beam equation with weak-internal damping in domain with moving boundary, Applied Numerical Mathematics, 47 (2003), 139-157.  doi: 10.1016/S0168-9274(03)00066-7.

[10]

H. R. Clark, Elastic membrane equation in bounded and unbounded domains, EJQTDE, 11 (2002), 1-21. 

[11]

R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl., 29 (1970), 443-454.  doi: 10.1016/0022-247X(70)90094-6.

[12]

E. H. Dowell, Aeroelasticity of Plates and Shells, Groninger, NL, Noordhoff Int. Publishing Co., 1975.

[13]

A. Eden and A. J. Milani, Exponential attractor for extensible beam equations, Nonlinearity, 6 (1993), 457-479.  doi: 10.1088/0951-7715/6/3/007.

[14]

C. Giorgi, M. G. Naso, V. Pata and M. Potomkin, Global attractors for the extensible thermoelastic beam system, J. Differential Equations, 246 (2009), 3496-3517.  doi: 10.1016/j.jde.2009.02.020.

[15]

T. J. Hughes and J. E. Marsden, Mathematical Foundation of Elasticity, Dover Publications, Inc., New York, 1994.

[16]

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. 

[17]

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.

[18]

M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.

[19]

H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. 

[20]

J. Limaco, H. R. Clark and A. J. Feitosa, Beam evolution equation with variable coeficients, Math. Meth. Appl. Sci., 28 (2005), 457-478.  doi: 10.1002/mma.577.

[21]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.

[22]

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.

[23]

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.

[24]

C. Mu and J. Ma, On a system of nonlinear wave equations with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 65 (2014), 91-113.  doi: 10.1007/s00033-013-0324-2.

[25]

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.

[26]

M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan, 30 (1978), 747-762.  doi: 10.2969/jmsj/03040747.

[27]

S. K. 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.

[28]

S. Woinowsky-Krieger, The effect of axial force on the vibration of hinged bars, Journal of Applied Mechanics, 17 (1950), 35-36. 

[29]

Y. You, Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping, Abstr. Appl. Anal., 1 (1996), 83-102.  doi: 10.1155/S1085337596000048.

[30]

W. Zhang, Nonlinear damping model: Response to random excitation, 5th Annual NASA Spacecraft Control Laboratory Experiment (SCOLE) Workshop, (1988), 27–38.

[31]

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, A theory of nonlinear damping in flexible structures, Stabilization of flexible structures, (1988), 1–12.

[2]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.

[3]

J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.

[4]

J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418.  doi: 10.1016/0022-0396(73)90056-9.

[5]

R. W. Bass and D. Zes, Spillover, Nonlinearity, and flexible structures, The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, NASA Conference Publication 10065 ed. L.W.Taylor, (1991), 1–14.

[6]

A. C. Biazutti and H. R. Crippa, Global attractor and inertial set for the beam equation, Applicable Analysis, 55 (1994), 61-78.  doi: 10.1080/00036819408840290.

[7]

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.

[8]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math., 195 (2008), ⅷ+183 pp. doi: 10.1090/memo/0912.

[9]

H. R. Clark, M. A. Rincon and R. D. Rodrigues, Beam equation with weak-internal damping in domain with moving boundary, Applied Numerical Mathematics, 47 (2003), 139-157.  doi: 10.1016/S0168-9274(03)00066-7.

[10]

H. R. Clark, Elastic membrane equation in bounded and unbounded domains, EJQTDE, 11 (2002), 1-21. 

[11]

R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl., 29 (1970), 443-454.  doi: 10.1016/0022-247X(70)90094-6.

[12]

E. H. Dowell, Aeroelasticity of Plates and Shells, Groninger, NL, Noordhoff Int. Publishing Co., 1975.

[13]

A. Eden and A. J. Milani, Exponential attractor for extensible beam equations, Nonlinearity, 6 (1993), 457-479.  doi: 10.1088/0951-7715/6/3/007.

[14]

C. Giorgi, M. G. Naso, V. Pata and M. Potomkin, Global attractors for the extensible thermoelastic beam system, J. Differential Equations, 246 (2009), 3496-3517.  doi: 10.1016/j.jde.2009.02.020.

[15]

T. J. Hughes and J. E. Marsden, Mathematical Foundation of Elasticity, Dover Publications, Inc., New York, 1994.

[16]

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. 

[17]

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.

[18]

M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.

[19]

H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. 

[20]

J. Limaco, H. R. Clark and A. J. Feitosa, Beam evolution equation with variable coeficients, Math. Meth. Appl. Sci., 28 (2005), 457-478.  doi: 10.1002/mma.577.

[21]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.

[22]

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.

[23]

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.

[24]

C. Mu and J. Ma, On a system of nonlinear wave equations with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 65 (2014), 91-113.  doi: 10.1007/s00033-013-0324-2.

[25]

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.

[26]

M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan, 30 (1978), 747-762.  doi: 10.2969/jmsj/03040747.

[27]

S. K. 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.

[28]

S. Woinowsky-Krieger, The effect of axial force on the vibration of hinged bars, Journal of Applied Mechanics, 17 (1950), 35-36. 

[29]

Y. You, Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping, Abstr. Appl. Anal., 1 (1996), 83-102.  doi: 10.1155/S1085337596000048.

[30]

W. Zhang, Nonlinear damping model: Response to random excitation, 5th Annual NASA Spacecraft Control Laboratory Experiment (SCOLE) Workshop, (1988), 27–38.

[31]

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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