July  2015, 35(7): 2881-2904. doi: 10.3934/dcds.2015.35.2881

Exponential attractors for abstract equations with memory and applications to viscoelasticity

1. 

Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano, Italy

2. 

Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara

3. 

Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Via Bonardi 9, Milano 20133

Received  June 2014 Revised  October 2014 Published  January 2015

We consider an abstract equation with memory of the form $$\partial_t x(t) + \int_{0}^\infty k(s) A x(t-s) ds + Bx(t)=0$$ where $A,B$ are operators acting on some Banach space, and the convolution kernel $k$ is a nonnegative convex summable function of unit mass. The system is translated into an ordinary differential equation on a Banach space accounting for the presence of memory, both in the so-called history space framework and in the minimal state one. The main theoretical result is a theorem providing sufficient conditions in order for the related solution semigroups to possess finite-dimensional exponential attractors. As an application, we prove the existence of exponential attractors for the integrodifferential equation $$\partial_{t t} u - h(0)\Delta u - \int_{0}^\infty h'(s) \Delta u(t-s) ds+ f(u) = g$$ arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel $h(s)=k(s)+1$.
Citation: Valeria Danese, Pelin G. Geredeli, Vittorino Pata. Exponential attractors for abstract equations with memory and applications to viscoelasticity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2881-2904. doi: 10.3934/dcds.2015.35.2881
References:
[1]

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

[2]

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

[3]

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

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I. Chueshov and I. Lasiecka, Long-Time Behavior of Second-Order Evolution Equations with Nonlinear Damping,, Amer. Math. Soc., (2008).  doi: 10.1090/memo/0912.  Google Scholar

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M. Conti, E. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework,, Discrete Contin. Dyn. Syst., 27 (2010), 1535.  doi: 10.3934/dcds.2010.27.1535.  Google Scholar

[6]

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

[7]

M. Conti, V. Pata and M. Squassina, Singular limit of dissipative hyperbolic equations with memory,, Discrete Contin. Dyn. Syst., (2005), 200.   Google Scholar

[8]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory,, Indiana Univ. Math. J., 55 (2006), 169.  doi: 10.1512/iumj.2006.55.2661.  Google Scholar

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C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rational Mech. Anal., 37 (1970), 297.   Google Scholar

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G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity,, Arch. Rational Mech. Anal., 138 (1997), 1.  doi: 10.1007/s002050050035.  Google Scholar

[11]

L. Deseri, M. Fabrizio and M.J. Golden, The concept of minimal state in viscoelasticity: New free energies an applications to PDEs,, Arch. Rational Mech. Anal., 181 (2006), 43.  doi: 10.1007/s00205-005-0406-1.  Google Scholar

[12]

F. Di Plinio and V. Pata, Robust exponential attractors for the strongly damped wave equation with memory. I,, Russian J. Math. Phys., 15 (2008), 301.  doi: 10.1134/S1061920808030014.  Google Scholar

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Masson, (1994).   Google Scholar

[14]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$,, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

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M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703.  doi: 10.1017/S030821050000408X.  Google Scholar

[16]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, Discrete Contin. Dyn. Syst., 10 (2004), 211.  doi: 10.3934/dcds.2004.10.211.  Google Scholar

[17]

M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory,, Arch. Rational Mech. Anal., 198 (2010), 189.  doi: 10.1007/s00205-010-0300-3.  Google Scholar

[18]

D. Fargue, Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles),, (French) [Reducibility of hereditary systems to dynamical systems], 277 (1973).   Google Scholar

[19]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of first order evolution equations,, Nonlinearity, 18 (2005), 1859.  doi: 10.1088/0951-7715/18/4/023.  Google Scholar

[20]

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.  doi: 10.1216/RMJ-2008-38-4-1117.  Google Scholar

[21]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329.  doi: 10.1017/S0308210509000365.  Google Scholar

[22]

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

[23]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, (French) [Dissipative dynamical systems and applications], (1991).   Google Scholar

[24]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV (eds. C.M. Dafermos and M. Pokorny), (2008), 103.  doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[25]

V. Pata, Exponential stability in linear viscoelasticity,, Quarterly of Applied Mathematics, 64 (2006), 499.  doi: 10.1007/s00032-009-0098-3.  Google Scholar

[26]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels,, Commun. Pure Appl. Anal., 9 (2010), 721.  doi: 10.3934/cpaa.2010.9.721.  Google Scholar

[27]

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

[28]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity,, Longman Scientific & Technical, (1987).   Google Scholar

[29]

J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge University Press, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar

[30]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1997).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second-Order Evolution Equations with Nonlinear Damping,, Amer. Math. Soc., (2008).  doi: 10.1090/memo/0912.  Google Scholar

[5]

M. Conti, E. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework,, Discrete Contin. Dyn. Syst., 27 (2010), 1535.  doi: 10.3934/dcds.2010.27.1535.  Google Scholar

[6]

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

[7]

M. Conti, V. Pata and M. Squassina, Singular limit of dissipative hyperbolic equations with memory,, Discrete Contin. Dyn. Syst., (2005), 200.   Google Scholar

[8]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory,, Indiana Univ. Math. J., 55 (2006), 169.  doi: 10.1512/iumj.2006.55.2661.  Google Scholar

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rational Mech. Anal., 37 (1970), 297.   Google Scholar

[10]

G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity,, Arch. Rational Mech. Anal., 138 (1997), 1.  doi: 10.1007/s002050050035.  Google Scholar

[11]

L. Deseri, M. Fabrizio and M.J. Golden, The concept of minimal state in viscoelasticity: New free energies an applications to PDEs,, Arch. Rational Mech. Anal., 181 (2006), 43.  doi: 10.1007/s00205-005-0406-1.  Google Scholar

[12]

F. Di Plinio and V. Pata, Robust exponential attractors for the strongly damped wave equation with memory. I,, Russian J. Math. Phys., 15 (2008), 301.  doi: 10.1134/S1061920808030014.  Google Scholar

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Masson, (1994).   Google Scholar

[14]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$,, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[15]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703.  doi: 10.1017/S030821050000408X.  Google Scholar

[16]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, Discrete Contin. Dyn. Syst., 10 (2004), 211.  doi: 10.3934/dcds.2004.10.211.  Google Scholar

[17]

M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory,, Arch. Rational Mech. Anal., 198 (2010), 189.  doi: 10.1007/s00205-010-0300-3.  Google Scholar

[18]

D. Fargue, Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles),, (French) [Reducibility of hereditary systems to dynamical systems], 277 (1973).   Google Scholar

[19]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of first order evolution equations,, Nonlinearity, 18 (2005), 1859.  doi: 10.1088/0951-7715/18/4/023.  Google Scholar

[20]

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.  doi: 10.1216/RMJ-2008-38-4-1117.  Google Scholar

[21]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329.  doi: 10.1017/S0308210509000365.  Google Scholar

[22]

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

[23]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, (French) [Dissipative dynamical systems and applications], (1991).   Google Scholar

[24]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV (eds. C.M. Dafermos and M. Pokorny), (2008), 103.  doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[25]

V. Pata, Exponential stability in linear viscoelasticity,, Quarterly of Applied Mathematics, 64 (2006), 499.  doi: 10.1007/s00032-009-0098-3.  Google Scholar

[26]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels,, Commun. Pure Appl. Anal., 9 (2010), 721.  doi: 10.3934/cpaa.2010.9.721.  Google Scholar

[27]

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

[28]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity,, Longman Scientific & Technical, (1987).   Google Scholar

[29]

J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge University Press, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar

[30]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1997).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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