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 & 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, (1992).   Google Scholar

[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.   Google Scholar

[3]

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

[4]

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

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors of Equations of Mathematical Physics,", American Mathematical Society Colloquium Publications, (2002).   Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.   Google Scholar

[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.  doi: 10.1216/RMJ-2008-38-4-1117.  Google Scholar

[15]

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

[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.   Google Scholar

[17]

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

[18]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory,, Evolution Equations, Semigroups and Functional Analysis, (2002), 155.   Google Scholar

[19]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs, (1988).   Google Scholar

[20]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Recherches en Mathématiques Appliqués [Research in Applied Mathematics], (1991).   Google Scholar

[21]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Cambridge University Press, (1991).   Google Scholar

[22]

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

[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.   Google Scholar

[24]

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

[25]

A. Miranville and S. Zelik, "Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,", Handbook of Differential Equations: Evolutionary Equations. Vol. \textbf{IV}, IV (2008), 103.   Google Scholar

[26]

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

[27]

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

[28]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, (1983).   Google Scholar

[29]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, (1997).   Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).   Google Scholar

[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.   Google Scholar

[3]

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

[4]

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

[5]

V. V. Chepyzhov and M. I. Vishik, "Attractors of Equations of Mathematical Physics,", American Mathematical Society Colloquium Publications, (2002).   Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.   Google Scholar

[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.  doi: 10.1216/RMJ-2008-38-4-1117.  Google Scholar

[15]

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

[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.   Google Scholar

[17]

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

[18]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory,, Evolution Equations, Semigroups and Functional Analysis, (2002), 155.   Google Scholar

[19]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs, (1988).   Google Scholar

[20]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Recherches en Mathématiques Appliqués [Research in Applied Mathematics], (1991).   Google Scholar

[21]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Cambridge University Press, (1991).   Google Scholar

[22]

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

[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.   Google Scholar

[24]

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

[25]

A. Miranville and S. Zelik, "Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,", Handbook of Differential Equations: Evolutionary Equations. Vol. \textbf{IV}, IV (2008), 103.   Google Scholar

[26]

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

[27]

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

[28]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, (1983).   Google Scholar

[29]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, (1997).   Google Scholar

[1]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[2]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[3]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147

[4]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024

[5]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105

[6]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[7]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

[8]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048

[9]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

[10]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[11]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[12]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

[13]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[14]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[15]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459

[16]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012

[17]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[18]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[19]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[20]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (2)

[Back to Top]