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Global attractors for nonlinear viscoelastic equations with memory
1. | Dipartimento di Matematica "F.Brioschi", Politecnico di Milano, I-20133 Milano |
2. | Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy |
References:
[1] |
R. O. Araujo, T. F. Ma and Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differential Equations, 254 (2013), 4066-4087.
doi: 10.1016/j.jde.2013.02.010. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[3] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053.
doi: 10.1002/mma.250. |
[4] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998. |
[5] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, 2002. |
[7] |
M. Conti, E. M. Marchini and V. Pata, A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94 (2014), 206-216.
doi: 10.1016/j.na.2013.08.015. |
[8] |
M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.
doi: 10.3934/cpaa.2005.4.705. |
[9] |
M. Conti and V. Pata, On the regularity of global attractors, Discrete Contin. Dyn. Syst., 25 (2009), 1209-1217.
doi: 10.3934/dcds.2009.25.1209. |
[10] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308. |
[11] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math., 38 (2008), 1117-1138.
doi: 10.1216/RMJ-2008-38-4-1117. |
[12] |
M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution Equations, Semigroups and Functional Analysis (A. Lorenzi and B. Ruf, Eds.) pp. 155-178, Progr. Nonlinear Differential Equations Appl. no. 50, Birkhäuser, Basel, 2002. |
[13] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. , Providence, 1988. |
[14] |
X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Methods Appl. Sci., 32 (2009), 346-358.
doi: 10.1002/mma.1041. |
[15] |
X. Han and M. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal., 70 (2009), 3090-3098.
doi: 10.1016/j.na.2008.04.011. |
[16] |
A. Haraux, Systèmes dynamiques dissipatifs et applications, Masson, Paris, 1991. |
[17] |
A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124.
doi: 10.1007/s005260050133. |
[18] |
W. Liu, Uniform decay of solutions for a quasilinear system of viscoelastic equations, Nonlinear Anal., 71 (2009), 2257-2267.
doi: 10.1016/j.na.2009.01.060. |
[19] |
W. Liu, General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source, Nonlinear Anal., 73 (2010), 1890-1904.
doi: 10.1016/j.na.2010.05.023. |
[20] |
A. H. Love, A Treatise on Mathematical Theory of Elasticity, Dover, New York, 1944. |
[21] |
S. A. Messaoudi and M. I. Mustafa, A general stability result for a quasilinear wave equation with memory, Nonlinear Anal. Real World Appl., 14 (2013), 1854-1864.
doi: 10.1016/j.nonrwa.2012.12.002. |
[22] |
S. A. Messaoudi and N. -e. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Methods Appl. Sci., 30 (2007), 665-680.
doi: 10.1002/mma.804. |
[23] |
S. A. Messaoudi and N. -e. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equation, Nonlinear Anal., 68 (2008), 785-793.
doi: 10.1016/j.na.2006.11.036. |
[24] |
S. A. Messaoudi and N. -e. Tatar, Exponential decay for a quasilinear viscoelastic equation, Math. Nachr., 282 (2009), 1443-1450.
doi: 10.1002/mana.200610800. |
[25] |
J. Y. Park and S. H. Park, General decay for quasilinear viscoelastic equations with nonlinear weak damping, J. Math. Phys., 50 (2009), 083505, 10 pp.
doi: 10.1063/1.3187780. |
[26] |
V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. |
[27] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[28] |
S. -T. Wu, Arbitrary decays for a viscoelastic equation, Bound. Value Probl., 28 (2011), 14 pp. |
show all references
References:
[1] |
R. O. Araujo, T. F. Ma and Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differential Equations, 254 (2013), 4066-4087.
doi: 10.1016/j.jde.2013.02.010. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[3] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053.
doi: 10.1002/mma.250. |
[4] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998. |
[5] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, 2002. |
[7] |
M. Conti, E. M. Marchini and V. Pata, A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94 (2014), 206-216.
doi: 10.1016/j.na.2013.08.015. |
[8] |
M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.
doi: 10.3934/cpaa.2005.4.705. |
[9] |
M. Conti and V. Pata, On the regularity of global attractors, Discrete Contin. Dyn. Syst., 25 (2009), 1209-1217.
doi: 10.3934/dcds.2009.25.1209. |
[10] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308. |
[11] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math., 38 (2008), 1117-1138.
doi: 10.1216/RMJ-2008-38-4-1117. |
[12] |
M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution Equations, Semigroups and Functional Analysis (A. Lorenzi and B. Ruf, Eds.) pp. 155-178, Progr. Nonlinear Differential Equations Appl. no. 50, Birkhäuser, Basel, 2002. |
[13] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. , Providence, 1988. |
[14] |
X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Methods Appl. Sci., 32 (2009), 346-358.
doi: 10.1002/mma.1041. |
[15] |
X. Han and M. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal., 70 (2009), 3090-3098.
doi: 10.1016/j.na.2008.04.011. |
[16] |
A. Haraux, Systèmes dynamiques dissipatifs et applications, Masson, Paris, 1991. |
[17] |
A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124.
doi: 10.1007/s005260050133. |
[18] |
W. Liu, Uniform decay of solutions for a quasilinear system of viscoelastic equations, Nonlinear Anal., 71 (2009), 2257-2267.
doi: 10.1016/j.na.2009.01.060. |
[19] |
W. Liu, General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source, Nonlinear Anal., 73 (2010), 1890-1904.
doi: 10.1016/j.na.2010.05.023. |
[20] |
A. H. Love, A Treatise on Mathematical Theory of Elasticity, Dover, New York, 1944. |
[21] |
S. A. Messaoudi and M. I. Mustafa, A general stability result for a quasilinear wave equation with memory, Nonlinear Anal. Real World Appl., 14 (2013), 1854-1864.
doi: 10.1016/j.nonrwa.2012.12.002. |
[22] |
S. A. Messaoudi and N. -e. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Methods Appl. Sci., 30 (2007), 665-680.
doi: 10.1002/mma.804. |
[23] |
S. A. Messaoudi and N. -e. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equation, Nonlinear Anal., 68 (2008), 785-793.
doi: 10.1016/j.na.2006.11.036. |
[24] |
S. A. Messaoudi and N. -e. Tatar, Exponential decay for a quasilinear viscoelastic equation, Math. Nachr., 282 (2009), 1443-1450.
doi: 10.1002/mana.200610800. |
[25] |
J. Y. Park and S. H. Park, General decay for quasilinear viscoelastic equations with nonlinear weak damping, J. Math. Phys., 50 (2009), 083505, 10 pp.
doi: 10.1063/1.3187780. |
[26] |
V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. |
[27] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[28] |
S. -T. Wu, Arbitrary decays for a viscoelastic equation, Bound. Value Probl., 28 (2011), 14 pp. |
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