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Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $
On a beam model related to flight structures with nonlocal energy damping
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 |
$ n $ |
$ 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}^+, $ |
$ \Omega\subset \mathbb{R}^n $ |
$ \kappa $ |
$ \gamma $ |
$ q\ge 1 $ |
$ f(u) $ |
References:
[1] |
A. V. Balakrishnan, A theory of nonlinear damping in flexible structures, Stabilization of flexible structures, (1988), 1–12. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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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