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May  2020, 40(5): 2561-2591. doi: 10.3934/dcds.2020141

Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations

Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

Received  November 2018 Revised  December 2019 Published  March 2020

In this paper we consider the nonlinear beam equations accounting for rotational inertial forces. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact global attractor and an exponential attractor. The main purpose is to trace the behavior of solutions of the nonlinear beam equations when the effect of the rotational inertia fades away gradually. A natural question is whether there are qualitative differences would appear or not. To answer the question, we deal with the rotational inertia with a parameter $ \alpha $ and consider the difference of behavior between the case $ 0<\alpha\le1 $ and the case $ \alpha = 0 $. The main novel contribution of this paper is to show the continuity of global attractors and exponential attractors with respect to $ \alpha $ in some sense.

Citation: Takayuki Niimura. Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2561-2591. doi: 10.3934/dcds.2020141
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland Publishing Co., Amsterdam, 1992.

[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]

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

[4]

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.

[5]

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

[6]

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.

[7]

P. Biler, Remark on the decay for damped string and beam equations, Nonlinear Anal., 10 (1986), 839-842.  doi: 10.1016/0362-546X(86)90071-4.

[8]

M. M. CavalcantiV. 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.

[9]

M. M. CavalcantiV. 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. 

[10]

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.

[11]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, ACTA, Kharkov, 1999,436 pp.

[12]

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

[13] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and LongTime Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.  doi: 10.1007/978-0-387-87712-9.
[14]

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.

[15]

E. H. de Brito, The damped elastic stretched string equation generalized: Existence, uniqueness, regularity, and stability, Applicable Anal., 13 (1982), 219-233.  doi: 10.1080/00036818208839392.

[16]

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.

[17]

R. W. Dickey, Dynamic stability of equilibrium states of the extensible beam, Proc. Amer. Math. Soc., 41 (1973), 94-102.  doi: 10.1090/S0002-9939-1973-0328290-8.

[18]

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.

[19]

A. EdenV. Kalantarov and A. Miranville, Finite-dimensional attractors for a general class of nonautonomous wave equations, Appl. Math. Lett., 13 (2000), 17-22.  doi: 10.1016/S0893-9659(00)00027-6.

[20]

M. Efendiev and A. Yagi, Continuous dependence on a parameter of exponential attractors for chemotaxis-growth syste, J. Math. Soc. Japan, 57 (2005), 167-181.  doi: 10.2969/jmsj/1160745820.

[21]

J. G. Eisley, Nonlinear vibration of beams and rectangular plates, Z. Angew. Math. Phys., 15 (1964), 167-175.  doi: 10.1007/BF01602658.

[22]

P. FabrieC. GalusinskiA. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.  doi: 10.3934/dcds.2004.10.211.

[23]

P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Analysis: Theory, Methods and Applications, 91 (2013), 72-92.  doi: 10.1016/j.na.2013.06.008.

[24]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[25]

J. S. HowellI. Lasiecka and J. T. Webster, Quasi-Stability and exponential attractors for a non-gradient system-applications to piston-theoretic plates with internal damping, Evollution Equations and Control Theory, 5 (2016), 567-603.  doi: 10.3934/eect.2016020.

[26]

J. S. HowellD Toundykov and J. T. Webster, A cantilevered extensible beam in axial flow: Semigroup well-posedness and postflutter regimes, SIAM Journal on Mathematical Analysis, 50 (2018), 2048-2085.  doi: 10.1137/17M1140261.

[27]

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.

[28]

S. Kouemou 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.

[29]

S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coeffcient, Nonlinear Anal., 71 (2009), 2361-2371.  doi: 10.1016/j.na.2009.01.187.

[30]

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

[31]

J. E. Lagnese and G. Leugering, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, J. Differential Equations, 91 (1991), 355-388. 

[32]

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

[33]

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.

[34]

T. F. MaV. 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.

[35]

L. A. Medeiros, On a new class of nonlinear wave equations, J. Math. Anal. Appl., 69 (1979), 252-262.  doi: 10.1016/0022-247X(79)90192-6.

[36]

C. L. 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.

[37]

J. E. Muñoz Rivera, Global solution and regularizing properties on a class of nonlinear evolution equation, J. Differential Equations, 128 (1996), 103-124.  doi: 10.1006/jdeq.1996.0091.

[38]

G. R. Sell and Y. C. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[39]

M. A. J. da 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.

[40]

M. A. Jorge SilvaV. Narciso and A. Vicent, On a beam model related to flight structures with nonlocal energy damping, Discrete Contin. Dyn. Syst. B, 24 (2019), 3281-3298. 

[41]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl.(4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[42]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[43]

C. F. Vasconcellos and L. M. Teixeira, Existence uniqueness and stabilization for a nonlinear plate system with nonlinear damping, Ann. Fac. Sci. Toulouse Math.(6), 8 (1999), 173-193.  doi: 10.5802/afst.928.

[44]

D. X. Wang and J. W. 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.

[45]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. 

[46]

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

[47]

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

[48]

Z. J. Yang, 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. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland Publishing Co., Amsterdam, 1992.

[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]

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

[4]

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.

[5]

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

[6]

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.

[7]

P. Biler, Remark on the decay for damped string and beam equations, Nonlinear Anal., 10 (1986), 839-842.  doi: 10.1016/0362-546X(86)90071-4.

[8]

M. M. CavalcantiV. 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.

[9]

M. M. CavalcantiV. 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. 

[10]

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.

[11]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, ACTA, Kharkov, 1999,436 pp.

[12]

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

[13] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and LongTime Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.  doi: 10.1007/978-0-387-87712-9.
[14]

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.

[15]

E. H. de Brito, The damped elastic stretched string equation generalized: Existence, uniqueness, regularity, and stability, Applicable Anal., 13 (1982), 219-233.  doi: 10.1080/00036818208839392.

[16]

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.

[17]

R. W. Dickey, Dynamic stability of equilibrium states of the extensible beam, Proc. Amer. Math. Soc., 41 (1973), 94-102.  doi: 10.1090/S0002-9939-1973-0328290-8.

[18]

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.

[19]

A. EdenV. Kalantarov and A. Miranville, Finite-dimensional attractors for a general class of nonautonomous wave equations, Appl. Math. Lett., 13 (2000), 17-22.  doi: 10.1016/S0893-9659(00)00027-6.

[20]

M. Efendiev and A. Yagi, Continuous dependence on a parameter of exponential attractors for chemotaxis-growth syste, J. Math. Soc. Japan, 57 (2005), 167-181.  doi: 10.2969/jmsj/1160745820.

[21]

J. G. Eisley, Nonlinear vibration of beams and rectangular plates, Z. Angew. Math. Phys., 15 (1964), 167-175.  doi: 10.1007/BF01602658.

[22]

P. FabrieC. GalusinskiA. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.  doi: 10.3934/dcds.2004.10.211.

[23]

P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Analysis: Theory, Methods and Applications, 91 (2013), 72-92.  doi: 10.1016/j.na.2013.06.008.

[24]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[25]

J. S. HowellI. Lasiecka and J. T. Webster, Quasi-Stability and exponential attractors for a non-gradient system-applications to piston-theoretic plates with internal damping, Evollution Equations and Control Theory, 5 (2016), 567-603.  doi: 10.3934/eect.2016020.

[26]

J. S. HowellD Toundykov and J. T. Webster, A cantilevered extensible beam in axial flow: Semigroup well-posedness and postflutter regimes, SIAM Journal on Mathematical Analysis, 50 (2018), 2048-2085.  doi: 10.1137/17M1140261.

[27]

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.

[28]

S. Kouemou 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.

[29]

S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coeffcient, Nonlinear Anal., 71 (2009), 2361-2371.  doi: 10.1016/j.na.2009.01.187.

[30]

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

[31]

J. E. Lagnese and G. Leugering, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, J. Differential Equations, 91 (1991), 355-388. 

[32]

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

[33]

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.

[34]

T. F. MaV. 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.

[35]

L. A. Medeiros, On a new class of nonlinear wave equations, J. Math. Anal. Appl., 69 (1979), 252-262.  doi: 10.1016/0022-247X(79)90192-6.

[36]

C. L. 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.

[37]

J. E. Muñoz Rivera, Global solution and regularizing properties on a class of nonlinear evolution equation, J. Differential Equations, 128 (1996), 103-124.  doi: 10.1006/jdeq.1996.0091.

[38]

G. R. Sell and Y. C. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[39]

M. A. J. da 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.

[40]

M. A. Jorge SilvaV. Narciso and A. Vicent, On a beam model related to flight structures with nonlocal energy damping, Discrete Contin. Dyn. Syst. B, 24 (2019), 3281-3298. 

[41]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl.(4), 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[42]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[43]

C. F. Vasconcellos and L. M. Teixeira, Existence uniqueness and stabilization for a nonlinear plate system with nonlinear damping, Ann. Fac. Sci. Toulouse Math.(6), 8 (1999), 173-193.  doi: 10.5802/afst.928.

[44]

D. X. Wang and J. W. 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.

[45]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. 

[46]

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

[47]

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

[48]

Z. J. Yang, 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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