Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Uniform attractors for a non-autonomous semilinear heat equation with memory

Authors: Claudio Giorgi, Vittorino Pata and Alfredo Marzocchi
Journal: Quart. Appl. Math. 58 (2000), 661-683
MSC: Primary 37L30; Secondary 35B41, 35K57, 35R10, 45K05
DOI: https://doi.org/10.1090/qam/1788423
MathSciNet review: MR1788423
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Abstract: In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of a non-autonomous integro-partial differential equation describing the heat flow in a rigid heat conductor with memory. Existence and uniqueness of solutions is provided. Moreover, under proper assumptions on the heat flux memory kernel and on the magnitude of nonlinearity, the existence of uniform absorbing sets and of a global uniform attractor is achieved. In the case of quasiperiodic dependence of time of the external heat supply, the above attractor is shown to have finite Hausdorff dimension.

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  • [1] S. Aizicovici and V. Barbu, Existence and asymptotic results for a system of integro-partial differential equations, NoDEA Nonlinear Differential Equations Appl. 3, 1-18 (1996) MR 1371092
  • [2] L. Amerio and G. Prouse, Abstract almost periodic functions and functional equations, Van Nostrand, New York, 1971 MR 0275061
  • [3] G. Bonfanti, P. Colli, M. Grasselli, and F. Luterotti, Nonsmooth kernels in a phase relaxation problem with memory, Nonlinear Anal. 32, 455-465 (1998) MR 1611150
  • [4] H. Brézis, Operateurs maximaux monotones, North-Holland, Amsterdam, 1973
  • [5] V. V. Chepyzhov and M. I. Vishik, Nonautonomous evolution equations and their attractors, Russian J. Math. Phys. 1, 165-190 (1993) MR 1259480
  • [6] V. V. Chepyzhov and M. I. Vishik, Attractors of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl. 73, 279-333 (1994) MR 1273705
  • [7] V. V. Chepyzhov and M. I. Vishik, Non-autonomous evolutionary equations with translation compact symbols and their attractor, C. R. Acad. Sci. Paris Sér. I Math. 321, 153-158 (1995) MR 1345438
  • [8] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equation for rigid heat conductors, Z. Angew. Math. Phys. 18, 199-208 (1967) MR 0214334
  • [9] P. Colli and M. Grasselli, Phase transition problems in materials with memory, J. Integral Equations Appl. 5, 1-22 (1993) MR 1214707
  • [10] P. Constantin, C. Foias, and R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53, no. 314 (1993) MR 776345
  • [11] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297-308 (1970) MR 0281400
  • [12] W. Desch and R. K. Miller, Exponential stabilization of Volterra integrodifferential equations in Hilbert spaces, J. Differential Equations 70, 366-389 (1987) MR 915494
  • [13] G. Gentili and C. Giorgi, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math. 51, 343-362 (1993) MR 1218373
  • [14] C. Giorgi, A. Marzocchi, and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl. 5, 333-354 (1998) MR 1638908
  • [15] 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 A-76, 119-137 (1976-77)
  • [16] M. Grasselli and V. Pata, Longtime behavior of a homogenized model in viscoelastodynamics, Discrete Contin. Dynam. Systems 4, 339-358 (1998) MR 1617314
  • [17] A. Haraux, Systèmes dynamiques dissipatifs et applications, Masson, Paris, 1990 MR 1084372
  • [18] R. K. Miller, An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66, 331-332 (1978) MR 515894
  • [19] J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29, 187-204 (1971) MR 0295683
  • [20] V. Pata, G. Prouse, and M. I. Vishik, Traveling waves of dissipative non-autonomous hyperbolic equations in a strip, Adv. Differential Equations 3, 249-270 (1998) MR 1750416
  • [21] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, submitted MR 1907454
  • [22] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer-Verlag, New York, 1988 MR 953967

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DOI: https://doi.org/10.1090/qam/1788423
Article copyright: © Copyright 2000 American Mathematical Society

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