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A Remark on Nonclassical Diffusion Equations with Memory

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Abstract

The nonclassical diffusion equation with hereditary memory

$$\begin{aligned} u_t-\Delta u_t-\Delta u-\int _0^\infty \kappa (s)\Delta u(t-s)\,\mathrm{d}s +\varphi (u)=f \end{aligned}$$

on a 3D bounded domain is considered, for a very general class of memory kernels \(\kappa \). Setting the problem both in the classical past history framework and in the more recent minimal state one, the related solution semigroups are shown to possess finite-dimensional regular exponential attractors.

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Notes

  1. Denoting by \(N(r)\) the smallest number of \(r\)-balls of \(\mathcal H\) necessary to cover \({\mathcal E}\), the fractal dimension of \({\mathcal E}\) in \(\mathcal H\) is defined as

    $$\begin{aligned} \limsup _{r\rightarrow 0} \frac{ \ln N(r)}{\ln \frac{1}{r}}. \end{aligned}$$

  2. Actually, \(\tilde{S}(t)\) maps \(\mathcal V^\sigma \) into \(\mathcal V^\sigma \) for every \(\sigma \in [0,1]\).

References

  1. Aifantis, E.C.: On the problem of diffusion in solids. Acta Mech. 37, 265–296 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland, Amsterdam (1992)

    MATH  Google Scholar 

  3. Chekroun, M.D., Di Plinio, F., Glatt-Holtz, N.E., Pata, V.: Asymptotics of the Coleman–Gurtin model. Discret. Contin. Dyn. Syst. Ser. S 4, 351–369 (2011)

    MATH  Google Scholar 

  4. Conti, M., Marchini, E.M., Pata, V.: Nonclassical diffusion with memory. Math. Methods Appl. Sci. (In press)

  5. Conti, M., Pata, V., Squassina, M.: Singular limit of differential systems with memory. Indiana Univ. Math. J. 55, 170–213 (2006)

    Article  MathSciNet  Google Scholar 

  6. Conti, M., Marchini, E.M., Pata, V.: Semilinear wave equations of viscoelasticity in the minimal state framework. Discret. Contin. Dyn. Syst 27, 1535–1552 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Conti, M., Marchini, E.M.: Wave equations with memory: the minimal state approach. J. Math. Anal. Appl. 384, 607–625 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Conti, M., Marchini, E.M., Pata, V.: Reaction-diffusion with memory in the minimal state framework. Trans. Am. Math. Soc 366, 4969–4986 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 554–569 (1970)

    Article  MathSciNet  Google Scholar 

  10. Danese, V., Geredeli, P.G., Pata, V.: Exponential attractors for abstract equations with memory and applications to viscoelasticity. Discret. Contin. Dyn. Syst. (In press)

  11. Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics, vol. 37. John-Wiley, New York (1994)

    Google Scholar 

  12. Efendiev, M., Miranville, A., Zelik, S.: Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems. Proc. Roy. Soc. Edinburgh Sect. A 13, 703–730 (2005)

    Article  MathSciNet  Google Scholar 

  13. Fabrie, P., Galusinski, C., Miranville, A., Zelik, S.: Uniform exponential attractors for a singularly perturbed damped wave equation. Discret. Contin. Dyn. Syst 10, 211–238 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fabrizio, M., Giorgi, C., Pata, V.: A new approach to equations with memory. Arch. Ration. Mech. Anal. 198, 189–232 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Grasselli, M., Pata, V.: Uniform attractors of nonautonomous systems with memory. In: Lorenzi, A., Ruf, B. (eds.) Evolution Equations, Semigroups and Functional Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol. 50, pp. 155–178. Birkhäuser, Boston (2002)

  16. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence (1988)

    MATH  Google Scholar 

  17. Jäckle, J.: Heat conduction and relaxation in liquids of high viscosity. Phys. A 162, 377–404 (1990)

    Article  Google Scholar 

  18. Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. In: Dafermos, C.M., Pokorny, M. (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. 4. Elsevier, Amsterdam (2008)

  19. Pan, L., Liu, Y.: Robust exponential attractors for the non-autonomous nonclassical diffusion equation with memory. Dyn. Syst. 28, 501–517 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sun, C., Yang, M.: Dynamics of the nonclassical diffusion equations. Asymptot. Anal. 59, 51–81 (2008)

    MATH  MathSciNet  Google Scholar 

  21. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  22. Wang, X., Zhong, C.: Attractors for the non-autonomous nonclassical diffusion equation with fading memory. Nonlinear Anal. 71, 5733–5746 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wang, X., Yang, L., Zhong, C.: Attractors for the nonclassical diffusion equation with fading memory. J. Math. Anal. Appl 362, 327–337 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133, Milano, Italy

    Monica Conti & Elsa M. Marchini

Authors
  1. Monica Conti
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  2. Elsa M. Marchini
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Correspondence to Monica Conti.

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The authors are members of the GNAMPA, Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni.

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Conti, M., Marchini, E.M. A Remark on Nonclassical Diffusion Equations with Memory. Appl Math Optim 73, 1–21 (2016). https://doi.org/10.1007/s00245-015-9290-8

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