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March  2015, 35(3): 985-1008. doi: 10.3934/dcds.2015.35.985

## Attractors and their properties for a class of nonlocal extensible beams

 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

Received  September 2013 Revised  August 2014 Published  October 2014

This paper is concerned with well-posedness and asymptotic behavior to a class of extensible beams with nonlinear fractional damping and source terms $$u_{tt} + \Delta^2 u - M\left(\int_{\Omega}|\nabla u|^2 \, dx \right)\Delta u + N\left(\int_{\Omega}|\nabla u|^2 \, dx \right) \left(-\Delta\right)^{\theta} u_t + f(u) = h$$ in $\Omega\times(0,\infty)$, where $\Omega\subset \mathbb{R}^q$ is a bounded domain with smooth boundary $\partial\Omega$, $0\leq \theta \le 1$, $M$ and $N$ are scalar functions, $f(u)$ is a source term and $h$ is a forcing term. When $\theta=1$ we have the classical strong damping $-\Delta u_t$, and if $\theta=0$ then weak damping $u_t$ is considered. Under the mathematical point of view the abstract setting of $\left(-\Delta\right)^{\theta}u_t$ with $0< \theta < 1$ constitutes a way to change the characteristic of the problem in wave and plate vibrations, and maybe the decay rate of solutions mainly when we consider nonconstant coefficient damping $N$. The main purpose in the present paper is to show that the transition from the case $0< \theta \leq 1$ to the case $\theta=0$ does not produce any influence on the asymptotic behavior of solutions, that is, all results are obtained by moving $\theta\in[0,1]$ uniformly. Our main result establishes the existence of finite-dimensional compact global and exponential attractors by using an approach on quasi-stable dynamical systems given by Chueshov and Lasiecka (2010).
Citation: Marcio Antonio Jorge da Silva, Vando Narciso. Attractors and their properties for a class of nonlocal extensible beams. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 985-1008. doi: 10.3934/dcds.2015.35.985
##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland, Amsterdam, 1992. [2] 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. [3] J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418. doi: 10.1016/0022-0396(73)90056-9. [4] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. [5] 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. [6] 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. [7] F. Bucci, I. Chueshov and I. Lasiecka, Global attractor for a composite system of nonlinear wave and plate equations, Commun. Pure Appl. Anal., 6 (2007), 113-140. doi: 10.3934/cpaa.2007.6.113. [8] F. Bucci and D. Toundykov, Finite-dimensional attractor for a composite system of wave/plate equations with localized damping, Nonlinearity, 23 (2010), 2271-2306. doi: 10.1088/0951-7715/23/9/011. [9] 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. [10] M. M. Cavalcanti, Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation, Discrete Contin. Dyn. Syst., 8 (2002), 675-695. doi: 10.3934/dcds.2002.8.675. [11] 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. [12] 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. [13] 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, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 66 (2006), 67-91. doi: 10.1007/3-7643-7401-2_5. [14] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002. [15] 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. [16] 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. [17] 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. [18] 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. [19] I. Chueshov and I. Lasiecka, Attractors for second order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x. [20] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 195, Providence, 2008. doi: 10.1090/memo/0912. [21] 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. [22] 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. [23] 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. [24] 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. [25] 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. [26] 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. [27] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Applied Mathematics, 37. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. [28] A. Eden, V. 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. [29] J. G. Eisley, Nonlinear vibration of beams and rectangular plates, Z. Angew. Math. Phys., 15 (1964), 167-175. doi: 10.1007/BF01602658. [30] C. Foias and E. Olson, Finite fractal dimension and Hölder-Lipschitz parametrization, Indiana Univ. Math. J., 45 (1996), 603-616. doi: 10.1512/iumj.1996.45.1326. [31] M. Grobbelaar-Van Dalsen and A. van der Merwe, Boundary stabilization for the extensible beam with attached load, Math. Models Methods Appl. Sci., 9 (1999), 379-394. doi: 10.1142/S0218202599000191. [32] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988. [33] 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. [34] 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. [35] 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. [36] G. Kirchhoff, Vorlessunger über Mathematiche Physik, Mechanik, Teubner, Leipzig, 1876. [37] 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. [38] 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. [39] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lezioni Lincee. [Lincei Lectures] Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. [40] H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. [41] P. Lazo, Global solutions for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601. doi: 10.1016/j.amc.2007.11.056. [42] T. F. Ma, Boundary stabilization for a non-linear beam on elastic bearings, Math. Methods Appl. Sci., 24 (2001), 583-594. doi: 10.1002/mma.230. [43] 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. [44] 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. [45] 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. [46] L. A. Medeiros and M. Milla Miranda, On a nonlinear wave equation with damping, Rev. Mat. Univ. Complut. Madrid, 3 (1990), 213-231. [47] 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. [48] A. F. Pazoto and G. Perla Menzala, Uniform Rates Of Decay of a Nonlinear Beam with Boundary Dissipation, Report of LNCC/CNPq (Brazil) no 34/97, August 1997. [49] A. F. Pazoto and G. Perla Menzala, Uniform rates of decay of a nonlinear beam model with thermal effects and nonlinear boundary dissipation, Funkcialaj Ekvacioj, 43 (2000), 339-360. [50] D. Ševčovič, Existence and limiting behaviour for damped nonlinear evolution equations with nonlocal terms, Comment. Math. Univ. Carolin., 31 (1990), 283-293. [51] 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. [52] 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. [53] 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. [54] 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. [55] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. [56] 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. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland, Amsterdam, 1992. [2] 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. [3] J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418. doi: 10.1016/0022-0396(73)90056-9. [4] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. [5] 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. [6] 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. [7] F. Bucci, I. Chueshov and I. Lasiecka, Global attractor for a composite system of nonlinear wave and plate equations, Commun. Pure Appl. Anal., 6 (2007), 113-140. doi: 10.3934/cpaa.2007.6.113. [8] F. Bucci and D. Toundykov, Finite-dimensional attractor for a composite system of wave/plate equations with localized damping, Nonlinearity, 23 (2010), 2271-2306. doi: 10.1088/0951-7715/23/9/011. [9] 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. [10] M. M. Cavalcanti, Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation, Discrete Contin. Dyn. Syst., 8 (2002), 675-695. doi: 10.3934/dcds.2002.8.675. [11] 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. [12] 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. [13] 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, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 66 (2006), 67-91. doi: 10.1007/3-7643-7401-2_5. [14] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002. [15] 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. [16] 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. [17] 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. [18] 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. [19] I. Chueshov and I. Lasiecka, Attractors for second order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x. [20] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 195, Providence, 2008. doi: 10.1090/memo/0912. [21] 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. [22] 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. [23] 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. [24] 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. [25] 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. [26] 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. [27] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Applied Mathematics, 37. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. [28] A. Eden, V. 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. [29] J. G. Eisley, Nonlinear vibration of beams and rectangular plates, Z. Angew. Math. Phys., 15 (1964), 167-175. doi: 10.1007/BF01602658. [30] C. Foias and E. Olson, Finite fractal dimension and Hölder-Lipschitz parametrization, Indiana Univ. Math. J., 45 (1996), 603-616. doi: 10.1512/iumj.1996.45.1326. [31] M. Grobbelaar-Van Dalsen and A. van der Merwe, Boundary stabilization for the extensible beam with attached load, Math. Models Methods Appl. Sci., 9 (1999), 379-394. doi: 10.1142/S0218202599000191. [32] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988. [33] 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. [34] 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. [35] 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. [36] G. Kirchhoff, Vorlessunger über Mathematiche Physik, Mechanik, Teubner, Leipzig, 1876. [37] 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. [38] 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. [39] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lezioni Lincee. [Lincei Lectures] Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. [40] H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. [41] P. Lazo, Global solutions for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601. doi: 10.1016/j.amc.2007.11.056. [42] T. F. Ma, Boundary stabilization for a non-linear beam on elastic bearings, Math. Methods Appl. Sci., 24 (2001), 583-594. doi: 10.1002/mma.230. [43] 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. [44] 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. [45] 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. [46] L. A. Medeiros and M. Milla Miranda, On a nonlinear wave equation with damping, Rev. Mat. Univ. Complut. Madrid, 3 (1990), 213-231. [47] 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. [48] A. F. Pazoto and G. Perla Menzala, Uniform Rates Of Decay of a Nonlinear Beam with Boundary Dissipation, Report of LNCC/CNPq (Brazil) no 34/97, August 1997. [49] A. F. Pazoto and G. Perla Menzala, Uniform rates of decay of a nonlinear beam model with thermal effects and nonlinear boundary dissipation, Funkcialaj Ekvacioj, 43 (2000), 339-360. [50] D. Ševčovič, Existence and limiting behaviour for damped nonlinear evolution equations with nonlocal terms, Comment. Math. Univ. Carolin., 31 (1990), 283-293. [51] 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. [52] 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. [53] 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. [54] 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. [55] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. [56] 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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