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September  2016, 15(5): 1893-1913. doi: 10.3934/cpaa.2016021

Global attractors for nonlinear viscoelastic equations with memory

Monica Conti 1, , Elsa M. Marchini 2, and  V. Pata 1,

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

Published  July 2016

We study the asymptotic properties of the semigroup $S(t)$ arising from the nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain \begin{eqnarray} |\partial_t u|^\rho \partial_{t t} u-\Delta \partial_{t t} u-\Delta \partial_t u\\ -\Big(1+\int_0^\infty \mu(s)\Delta s \Big)\Delta u +\int_0^\infty \mu(s)\Delta u(t-s)\Delta s +f(u)=h \end{eqnarray} written in the past history framework of Dafermos [10]. We establish the existence of the global attractor of optimal regularity for $S(t)$ when $\rho\in [0,4)$ and $f$ has polynomial growth of (at most) critical order 5.
Keywords: solution semigroup, global attractor., memory kernel, Nonlinear viscoelastic equations.
Mathematics Subject Classification: Primary: 35B41, 45G10; Secondary: 35L7.
Citation: Monica Conti, Elsa M. Marchini, V. Pata. Global attractors for nonlinear viscoelastic equations with memory. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1893-1913. doi: 10.3934/cpaa.2016021
References:
[1]

R. O. Araujo, T. F. Ma and Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history,, \emph{J. Differential Equations}, 254 (2013), 4066.  doi: 10.1016/j.jde.2013.02.010.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).   Google Scholar

[3]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping,, \emph{Math. Methods Appl. Sci.}, 24 (2001), 1043.  doi: 10.1002/mma.250.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998).   Google Scholar

[5]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity,, \emph{Asymptot. Anal.}, 46 (2006), 251.   Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002).   Google Scholar

[7]

M. Conti, E. M. Marchini and V. Pata, A well posedness result for nonlinear viscoelastic equations with memory,, \emph{Nonlinear Anal.}, 94 (2014), 206.  doi: 10.1016/j.na.2013.08.015.  Google Scholar

[8]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 705.  doi: 10.3934/cpaa.2005.4.705.  Google Scholar

[9]

M. Conti and V. Pata, On the regularity of global attractors,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 1209.  doi: 10.3934/dcds.2009.25.1209.  Google Scholar

[10]

C. M. Dafermos, Asymptotic stability in viscoelasticity,, \emph{Arch. Ration. Mech. Anal.}, 37 (1970), 297.   Google Scholar

[11]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation,, \emph{Rocky Mountain J. Math.}, 38 (2008), 1117.  doi: 10.1216/RMJ-2008-38-4-1117.  Google Scholar

[12]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory,, in \emph{Evolution Equations, (2002), 155.   Google Scholar

[13]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Amer. Math. Soc., (1988).   Google Scholar

[14]

X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping,, \emph{Math. Methods Appl. Sci.}, 32 (2009), 346.  doi: 10.1002/mma.1041.  Google Scholar

[15]

X. Han and M. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping,, \emph{Nonlinear Anal.}, 70 (2009), 3090.  doi: 10.1016/j.na.2008.04.011.  Google Scholar

[16]

A. Haraux, Systèmes dynamiques dissipatifs et applications,, Masson, (1991).   Google Scholar

[17]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity,, \emph{Calc. Var. Partial Differential Equations}, 9 (1999), 95.  doi: 10.1007/s005260050133.  Google Scholar

[18]

W. Liu, Uniform decay of solutions for a quasilinear system of viscoelastic equations,, \emph{Nonlinear Anal.}, 71 (2009), 2257.  doi: 10.1016/j.na.2009.01.060.  Google Scholar

[19]

W. Liu, General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source,, \emph{Nonlinear Anal.}, 73 (2010), 1890.  doi: 10.1016/j.na.2010.05.023.  Google Scholar

[20]

A. H. Love, A Treatise on Mathematical Theory of Elasticity,, Dover, (1944).   Google Scholar

[21]

S. A. Messaoudi and M. I. Mustafa, A general stability result for a quasilinear wave equation with memory,, \emph{Nonlinear Anal. Real World Appl.}, 14 (2013), 1854.  doi: 10.1016/j.nonrwa.2012.12.002.  Google Scholar

[22]

S. A. Messaoudi and N. -e. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem,, \emph{Math. Methods Appl. Sci.}, 30 (2007), 665.  doi: 10.1002/mma.804.  Google Scholar

[23]

S. A. Messaoudi and N. -e. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equation,, \emph{Nonlinear Anal.}, 68 (2008), 785.  doi: 10.1016/j.na.2006.11.036.  Google Scholar

[24]

S. A. Messaoudi and N. -e. Tatar, Exponential decay for a quasilinear viscoelastic equation,, \emph{Math. Nachr.}, 282 (2009), 1443.  doi: 10.1002/mana.200610800.  Google Scholar

[25]

J. Y. Park and S. H. Park, General decay for quasilinear viscoelastic equations with nonlinear weak damping,, \emph{J. Math. Phys.}, 50 (2009).  doi: 10.1063/1.3187780.  Google Scholar

[26]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory,, \emph{Adv. Math. Sci. Appl.}, 11 (2001), 505.   Google Scholar

[27]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[28]

S. -T. Wu, Arbitrary decays for a viscoelastic equation,, \emph{Bound. Value Probl.}, 28 (2011).   Google Scholar

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,, \emph{J. Differential Equations}, 254 (2013), 4066.  doi: 10.1016/j.jde.2013.02.010.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).   Google Scholar

[3]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping,, \emph{Math. Methods Appl. Sci.}, 24 (2001), 1043.  doi: 10.1002/mma.250.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998).   Google Scholar

[5]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity,, \emph{Asymptot. Anal.}, 46 (2006), 251.   Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002).   Google Scholar

[7]

M. Conti, E. M. Marchini and V. Pata, A well posedness result for nonlinear viscoelastic equations with memory,, \emph{Nonlinear Anal.}, 94 (2014), 206.  doi: 10.1016/j.na.2013.08.015.  Google Scholar

[8]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 705.  doi: 10.3934/cpaa.2005.4.705.  Google Scholar

[9]

M. Conti and V. Pata, On the regularity of global attractors,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 1209.  doi: 10.3934/dcds.2009.25.1209.  Google Scholar

[10]

C. M. Dafermos, Asymptotic stability in viscoelasticity,, \emph{Arch. Ration. Mech. Anal.}, 37 (1970), 297.   Google Scholar

[11]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation,, \emph{Rocky Mountain J. Math.}, 38 (2008), 1117.  doi: 10.1216/RMJ-2008-38-4-1117.  Google Scholar

[12]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory,, in \emph{Evolution Equations, (2002), 155.   Google Scholar

[13]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Amer. Math. Soc., (1988).   Google Scholar

[14]

X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping,, \emph{Math. Methods Appl. Sci.}, 32 (2009), 346.  doi: 10.1002/mma.1041.  Google Scholar

[15]

X. Han and M. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping,, \emph{Nonlinear Anal.}, 70 (2009), 3090.  doi: 10.1016/j.na.2008.04.011.  Google Scholar

[16]

A. Haraux, Systèmes dynamiques dissipatifs et applications,, Masson, (1991).   Google Scholar

[17]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity,, \emph{Calc. Var. Partial Differential Equations}, 9 (1999), 95.  doi: 10.1007/s005260050133.  Google Scholar

[18]

W. Liu, Uniform decay of solutions for a quasilinear system of viscoelastic equations,, \emph{Nonlinear Anal.}, 71 (2009), 2257.  doi: 10.1016/j.na.2009.01.060.  Google Scholar

[19]

W. Liu, General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source,, \emph{Nonlinear Anal.}, 73 (2010), 1890.  doi: 10.1016/j.na.2010.05.023.  Google Scholar

[20]

A. H. Love, A Treatise on Mathematical Theory of Elasticity,, Dover, (1944).   Google Scholar

[21]

S. A. Messaoudi and M. I. Mustafa, A general stability result for a quasilinear wave equation with memory,, \emph{Nonlinear Anal. Real World Appl.}, 14 (2013), 1854.  doi: 10.1016/j.nonrwa.2012.12.002.  Google Scholar

[22]

S. A. Messaoudi and N. -e. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem,, \emph{Math. Methods Appl. Sci.}, 30 (2007), 665.  doi: 10.1002/mma.804.  Google Scholar

[23]

S. A. Messaoudi and N. -e. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equation,, \emph{Nonlinear Anal.}, 68 (2008), 785.  doi: 10.1016/j.na.2006.11.036.  Google Scholar

[24]

S. A. Messaoudi and N. -e. Tatar, Exponential decay for a quasilinear viscoelastic equation,, \emph{Math. Nachr.}, 282 (2009), 1443.  doi: 10.1002/mana.200610800.  Google Scholar

[25]

J. Y. Park and S. H. Park, General decay for quasilinear viscoelastic equations with nonlinear weak damping,, \emph{J. Math. Phys.}, 50 (2009).  doi: 10.1063/1.3187780.  Google Scholar

[26]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory,, \emph{Adv. Math. Sci. Appl.}, 11 (2001), 505.   Google Scholar

[27]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[28]

S. -T. Wu, Arbitrary decays for a viscoelastic equation,, \emph{Bound. Value Probl.}, 28 (2011).   Google Scholar

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