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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 & Continuous Dynamical Systems - A, 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.  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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.   Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

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

[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.  Google Scholar

[47]

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

[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.  Google Scholar

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.  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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.   Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

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

[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.  Google Scholar

[47]

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

[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.  Google Scholar

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