-
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
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 |
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 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]