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Exponential attractors for abstract equations with memory and applications to viscoelasticity

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  • 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 $h(s)=k(s)+1$.
    Mathematics Subject Classification: Primary: 35B41, 37L30, 45K05; Secondary: 74D99.


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  • [1]

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


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


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


    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.


    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.


    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.


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


    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.


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


    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.


    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.


    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.


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


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


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


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


    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.


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


    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.


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


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


    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.


    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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