April  2011, 4(2): 351-369. doi: 10.3934/dcdss.2011.4.351

Asymptotics of the Coleman-Gurtin model

1. 

École Normale Supérieure - CERES-ERTI, Normale Supérieure - Ce 75231 Paris Cedex 05, France

2. 

Indiana University Mathematics Department and The Institute of Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, United States

3. 

Department of Mathematics and The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405

4. 

Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano

Received  January 2009 Revised  May 2009 Published  November 2010

This paper is concerned with the integrodifferential equation

$\partial_{t} u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\d s + \varphi(u)=f$

arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity $\varphi$ of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.

Citation: Mickaël D. Chekroun, Francesco di Plinio, Nathan Glatt-Holtz, Vittorino Pata. Asymptotics of the Coleman-Gurtin model. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 351-369. doi: 10.3934/dcdss.2011.4.351
References:
[1]

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

[2]

V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal., 50 (2006), 269-291.

[3]

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

[4]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), 351-370. doi: 10.1088/0951-7715/22/2/006.

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors of Equations of Mathematical Physics," American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002

[6]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208. doi: 10.1007/BF01596912.

[7]

M. Conti, S. Gatti, M. Grasselli and V. Pata, Two-dimensional reaction-diffusion equations with memory, Quart. Appl. Math., (in press).

[8]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 170-213. doi: 10.1512/iumj.2006.55.2661.

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 554-569. doi: 10.1007/BF00251609.

[10]

R. Datko, Extending a theorem of A. M. Liapunov to Hilbert space, J. Math. Anal. Appl., 32 (1970), 610-616. doi: 10.1016/0022-247X(70)90283-0.

[11]

F. Di Plinio, V. Pata and S. Zelik, On the strongly damped wave equation with memory, Indiana Univ. Math. J., 57 (2008), 757-780. doi: 10.1512/iumj.2008.57.3266.

[12]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $mathbb{R}^{3}$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[13]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 221-238.

[14]

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.

[15]

G. Gentili and C. Giorgi, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math., 51 (1993), 342-362.

[16]

C. Giorgi, A. Marzocchi and V. Pata, Uniform attractors for a non-autonomous semilinear heat equation with memory, Quart. Appl. Math., 58 (2000), 661-683.

[17]

H. Grabmüller, On linear theory of heat conduction in materials with memory, Proc. Roy. Soc. Edinburgh Sect. A, 76 (1976-77), 119-137.

[18]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, "Evolution Equations, Semigroups and Functional Analysis" (A. Lorenzi and B. Ruf, Eds.), 155-178, Progr. Nonlinear Differential Equations Appl. no.50, Birkhäuser, Basel, 2002.

[19]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[20]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliqués [Research in Applied Mathematics], 17, Masson, Paris, 1991.

[21]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Cambridge University Press, Cambridge, UK, 1991.

[22]

S. O. Londen and J. A. Nohel, Nonlinear Volterra integrodifferential equation occurring in heat flow, J. Integral Equations, 6 (1984), 11-50.

[23]

Y. I. Lysikov, On the possibility of development of vibrations during heating of the transparent dielectric by optical radiation, Zh. Prikl. Math. i Tekh. Fiz., 4 (1984), 56-59.

[24]

R. K. Miller, An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl., 66 (1978), 331-332. doi: 10.1016/0022-247X(78)90234-2.

[25]

A. Miranville and S. Zelik, "Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains," Handbook of Differential Equations: Evolutionary Equations. Vol. IV, 103-200, Elsevier/North-Holland, Amsterdam, 2008.

[26]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.

[27]

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

[28]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

[29]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

show all references

References:
[1]

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

[2]

V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal., 50 (2006), 269-291.

[3]

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

[4]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), 351-370. doi: 10.1088/0951-7715/22/2/006.

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors of Equations of Mathematical Physics," American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002

[6]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208. doi: 10.1007/BF01596912.

[7]

M. Conti, S. Gatti, M. Grasselli and V. Pata, Two-dimensional reaction-diffusion equations with memory, Quart. Appl. Math., (in press).

[8]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 170-213. doi: 10.1512/iumj.2006.55.2661.

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 554-569. doi: 10.1007/BF00251609.

[10]

R. Datko, Extending a theorem of A. M. Liapunov to Hilbert space, J. Math. Anal. Appl., 32 (1970), 610-616. doi: 10.1016/0022-247X(70)90283-0.

[11]

F. Di Plinio, V. Pata and S. Zelik, On the strongly damped wave equation with memory, Indiana Univ. Math. J., 57 (2008), 757-780. doi: 10.1512/iumj.2008.57.3266.

[12]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $mathbb{R}^{3}$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7.

[13]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 221-238.

[14]

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.

[15]

G. Gentili and C. Giorgi, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math., 51 (1993), 342-362.

[16]

C. Giorgi, A. Marzocchi and V. Pata, Uniform attractors for a non-autonomous semilinear heat equation with memory, Quart. Appl. Math., 58 (2000), 661-683.

[17]

H. Grabmüller, On linear theory of heat conduction in materials with memory, Proc. Roy. Soc. Edinburgh Sect. A, 76 (1976-77), 119-137.

[18]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, "Evolution Equations, Semigroups and Functional Analysis" (A. Lorenzi and B. Ruf, Eds.), 155-178, Progr. Nonlinear Differential Equations Appl. no.50, Birkhäuser, Basel, 2002.

[19]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[20]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliqués [Research in Applied Mathematics], 17, Masson, Paris, 1991.

[21]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Cambridge University Press, Cambridge, UK, 1991.

[22]

S. O. Londen and J. A. Nohel, Nonlinear Volterra integrodifferential equation occurring in heat flow, J. Integral Equations, 6 (1984), 11-50.

[23]

Y. I. Lysikov, On the possibility of development of vibrations during heating of the transparent dielectric by optical radiation, Zh. Prikl. Math. i Tekh. Fiz., 4 (1984), 56-59.

[24]

R. K. Miller, An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl., 66 (1978), 331-332. doi: 10.1016/0022-247X(78)90234-2.

[25]

A. Miranville and S. Zelik, "Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains," Handbook of Differential Equations: Evolutionary Equations. Vol. IV, 103-200, Elsevier/North-Holland, Amsterdam, 2008.

[26]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.

[27]

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

[28]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

[29]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

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