Attractors and their properties for a class of nonlocal extensible beams

Previous Article
On the quenching behaviour of a semilinear wave equation modelling MEMS technology
DCDS Home
This Issue
Next Article
Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one

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

Attractors and their properties for a class of nonlocal extensible beams

Marcio Antonio Jorge da Silva ^1, and Vando Narciso ^2,

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

Abstract
Related Papers

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).

Keywords: global attractor, Berger plate, long-time dynamics, fractal exponential attractor., Extensible beam.

Mathematics Subject Classification: Primary: 35B40, 35B41, 37L30; Secondary: 74H4.

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.

[1]	Marcio Antonio Jorge da Silva, Vando Narciso. Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping^. Evolution Equations and Control Theory*, 2017, 6 (3) : 437-470. doi: 10.3934/eect.2017023
[2]	George Avalos, Pelin G. Geredeli, Justin T. Webster. Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2301-2328. doi: 10.3934/cpaa.2016038
[3]	Igor Chueshov, Stanislav Kolbasin. Long-time dynamics in plate models with strong nonlinear damping. Communications on Pure and Applied Analysis, 2012, 11 (2) : 659-674. doi: 10.3934/cpaa.2012.11.659
[4]	Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041
[5]	Chunxiang Zhao, Chunyan Zhao, Chengkui Zhong. The global attractor for a class of extensible beams with nonlocal weak damping. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 935-955. doi: 10.3934/dcdsb.2019197
[6]	I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635
[7]	Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557
[8]	Yang Liu. Long-time behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28 (1) : 311-326. doi: 10.3934/era.2020018
[9]	Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure and Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113
[10]	Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151
[11]	Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations and Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241
[12]	Pelin G. Geredeli, Azer Khanmamedov. Long-time dynamics of the parabolic $p$-Laplacian equation. Communications on Pure and Applied Analysis, 2013, 12 (2) : 735-754. doi: 10.3934/cpaa.2013.12.735
[13]	Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2201-2238. doi: 10.3934/dcdsb.2020360
[14]	Tristan Roget. On the long-time behaviour of age and trait structured population dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2551-2576. doi: 10.3934/dcdsb.2018265
[15]	Igor Chueshov, Irena Lasiecka, Justin Webster. Flow-plate interactions: Well-posedness and long-time behavior. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 925-965. doi: 10.3934/dcdss.2014.7.925
[16]	Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055
[17]	Tingting Liu, Qiaozhen Ma, Ling Xu. Attractor of the Kirchhoff type plate equation with memory and nonlinear damping on the whole time-dependent space. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022046
[18]	Mykhailo Potomkin. Asymptotic behavior of thermoviscoelastic Berger plate. Communications on Pure and Applied Analysis, 2010, 9 (1) : 161-192. doi: 10.3934/cpaa.2010.9.161
[19]	Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717
[20]	Messoud Efendiev, Anna Zhigun. On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 651-673. doi: 10.3934/dcds.2018028

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Marcio Antonio Jorge da Silva Vando Narciso
on Google Scholar
Marcio Antonio Jorge da Silva Vando Narciso

[Back to Top]