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 and Continuous Dynamical Systems, 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, Amsterdam, 1992.

[2]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273.

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, 2002.

[4]

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

[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), 1535-1552. doi: 10.3934/dcds.2010.27.1535.

[6]

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.

[7]

M. Conti, V. Pata and M. Squassina, Singular limit of dissipative hyperbolic equations with memory, Discrete Contin. Dyn. Syst., Suppl., (2005), 200-208.

[8]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215. doi: 10.1512/iumj.2006.55.2661.

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.

[10]

G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal., 138 (1997), 1-35. doi: 10.1007/s002050050035.

[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), 43-96. doi: 10.1007/s00205-005-0406-1.

[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), 301-315. doi: 10.1134/S1061920808030014.

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994.

[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), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[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), 703-730. doi: 10.1017/S030821050000408X.

[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), 211-238. doi: 10.3934/dcds.2004.10.211.

[17]

M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Rational Mech. Anal., 198 (2010), 189-232. doi: 10.1007/s00205-010-0300-3.

[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], C.R. Acad. Sci. Paris Sér. B, 277 (1973), B471-B473.

[19]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of first order evolution equations, Nonlinearity, 18 (2005), 1859-1883. doi: 10.1088/0951-7715/18/4/023.

[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), 1117-1138. doi: 10.1216/RMJ-2008-38-4-1117.

[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), 329-366. doi: 10.1017/S0308210509000365.

[22]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, 1988.

[23]

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

[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), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[25]

V. Pata, Exponential stability in linear viscoelasticity, Quarterly of Applied Mathematics, 64 (2006), 499-513. doi: 10.1007/s00032-009-0098-3.

[26]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730. doi: 10.3934/cpaa.2010.9.721.

[27]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.

[28]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific & Technical, Harlow John Wiley & Sons, Inc., New York, 1987.

[29]

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

[30]

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

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[2]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273.

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, 2002.

[4]

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

[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), 1535-1552. doi: 10.3934/dcds.2010.27.1535.

[6]

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.

[7]

M. Conti, V. Pata and M. Squassina, Singular limit of dissipative hyperbolic equations with memory, Discrete Contin. Dyn. Syst., Suppl., (2005), 200-208.

[8]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215. doi: 10.1512/iumj.2006.55.2661.

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.

[10]

G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal., 138 (1997), 1-35. doi: 10.1007/s002050050035.

[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), 43-96. doi: 10.1007/s00205-005-0406-1.

[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), 301-315. doi: 10.1134/S1061920808030014.

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994.

[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), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[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), 703-730. doi: 10.1017/S030821050000408X.

[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), 211-238. doi: 10.3934/dcds.2004.10.211.

[17]

M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Rational Mech. Anal., 198 (2010), 189-232. doi: 10.1007/s00205-010-0300-3.

[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], C.R. Acad. Sci. Paris Sér. B, 277 (1973), B471-B473.

[19]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of first order evolution equations, Nonlinearity, 18 (2005), 1859-1883. doi: 10.1088/0951-7715/18/4/023.

[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), 1117-1138. doi: 10.1216/RMJ-2008-38-4-1117.

[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), 329-366. doi: 10.1017/S0308210509000365.

[22]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, 1988.

[23]

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

[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), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[25]

V. Pata, Exponential stability in linear viscoelasticity, Quarterly of Applied Mathematics, 64 (2006), 499-513. doi: 10.1007/s00032-009-0098-3.

[26]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730. doi: 10.3934/cpaa.2010.9.721.

[27]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.

[28]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific & Technical, Harlow John Wiley & Sons, Inc., New York, 1987.

[29]

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

[30]

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

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