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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Reaction-diffusion with memory in the minimal state framework


Authors: Monica Conti, Elsa M. Marchini and Vittorino Pata
Journal: Trans. Amer. Math. Soc. 366 (2014), 4969-4986
MSC (2010): Primary 35B41, 35K05, 45K05, 47H20
DOI: https://doi.org/10.1090/S0002-9947-2013-06097-7
Published electronically: November 6, 2013
MathSciNet review: 3217706
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the integrodifferential equation

$\displaystyle \partial _t u-\Delta u -\int _0^\infty \kappa (s)\Delta u(t-s)\,\textup {d} s + \varphi (u)=f$

arising in the Coleman-Gurtin theory of heat conduction with hereditary memory. Within a novel abstract framework, based on the notion of minimal state, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related semigroup of solutions.

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  • [1] A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. MR 1156492 (93d:58090)
  • [2] Mickaël D. Chekroun, Francesco Di Plinio, Nathan E. Glatt-Holtz, and Vittorino Pata, Asymptotics of the Coleman-Gurtin model, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), no. 2, 351-369. MR 2746378 (2012b:35164), https://doi.org/10.3934/dcdss.2011.4.351
  • [3] V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal. 46 (2006), no. 3-4, 251-273. MR 2215885 (2007c:47053)
  • [4] Vladimir V. Chepyzhov and Mark I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002. MR 1868930 (2003f:37001c)
  • [5] Bernard D. Coleman and Morton E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18 (1967), 199-208 (English, with German summary). MR 0214334 (35 #5185)
  • [6] Monica Conti and Elsa M. Marchini, Wave equations with memory: the minimal state approach, J. Math. Anal. Appl. 384 (2011), no. 2, 607-625. MR 2825211 (2012h:35224), https://doi.org/10.1016/j.jmaa.2011.06.009
  • [7] Monica Conti, Elsa M. Marchini, and Vittorino Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst. 27 (2010), no. 4, 1535-1552. MR 2629536 (2011c:35059), https://doi.org/10.3934/dcds.2010.27.1535
  • [8] Monica Conti, Vittorino Pata, and Marco Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J. 55 (2006), no. 1, 169-215. MR 2207550 (2006k:35290), https://doi.org/10.1512/iumj.2006.55.2661
  • [9] Constantine M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297-308. MR 0281400 (43 #7117)
  • [10] Gianpietro Del Piero and Luca Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal. 138 (1997), no. 1, 1-35. MR 1463802 (98i:73023), https://doi.org/10.1007/s002050050035
  • [11] Luca Deseri, Mauro Fabrizio, and Murrough Golden, The concept of minimal state in viscoelasticity: new free energies and applications to PDEs, Arch. Ration. Mech. Anal. 181 (2006), no. 1, 43-96. MR 2221203 (2009a:74024), https://doi.org/10.1007/s00205-005-0406-1
  • [12] Claudio Giorgi, Vittorino Pata, and Alfredo Marzocchi, Uniform attractors for a non-autonomous semilinear heat equation with memory, Quart. Appl. Math. 58 (2000), no. 4, 661-683. MR 1788423 (2001k:37129)
  • [13] H. Grabmüller, On linear theory of heat conduction in materials with memory. Existence and uniqueness theorems for the final value problem, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 2, 119-137. MR 0446112 (56 #4444)
  • [14] Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371 (89g:58059)
  • [15] Alain Haraux, Systèmes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 17, Masson, Paris, 1991 (French). MR 1084372 (92b:35002)
  • [16] Mauro Fabrizio, Claudio Giorgi, and Vittorino Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal. 198 (2010), no. 1, 189-232. MR 2679371 (2011e:35393), https://doi.org/10.1007/s00205-010-0300-3
  • [17] S.-O. Londen and J. A. Nohel, Nonlinear Volterra integro-differential equation occurring in heat flow, J. Integral Equations 6 (1984), no. 1, 11-50. MR 727934 (85j:45027)
  • [18] R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), no. 2, 313-332. MR 515894 (80g:45015), https://doi.org/10.1016/0022-247X(78)90234-2
  • [19] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of differential equations: evolutionary equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 103-200. MR 2508165 (2010c:37175), https://doi.org/10.1016/S1874-5717(08)00003-0
  • [20] Jace W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187-204. MR 0295683 (45 #4749)
  • [21] Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967 (89m:58056)

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Additional Information

Monica Conti
Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
Email: monica.conti@polimi.it

Elsa M. Marchini
Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
Email: elsa.marchini@polimi.it

Vittorino Pata
Affiliation: Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
Email: vittorino.pata@polimi.it

DOI: https://doi.org/10.1090/S0002-9947-2013-06097-7
Keywords: Heat conduction with memory, minimal state framework, solution semigroup, global attractor, exponential attractor
Received by editor(s): October 17, 2011
Received by editor(s) in revised form: February 1, 2013
Published electronically: November 6, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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