Rocky Mountain Journal of Mathematics

Attractors for Semi-Linear Equations of Viscoelasticity with Very Low Dissipation

S. Gatti, A. Miranville, V. Pata, and S. Zelik

Full-text: Open access

Article information

Rocky Mountain J. Math., Volume 38, Number 4 (2008), 1117-1138.

First available in Project Euclid: 1 July 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B40: Asymptotic behavior of solutions 35L70: Nonlinear second-order hyperbolic equations 37L45: Hyperbolicity; Lyapunov functions 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 74D99: None of the above, but in this section

Hyperbolic equation with memory dynamical system Lyapunov function gradient system global attractor


Gatti, S.; Miranville, A.; Pata, V.; Zelik, S. Attractors for Semi-Linear Equations of Viscoelasticity with Very Low Dissipation. Rocky Mountain J. Math. 38 (2008), no. 4, 1117--1138. doi:10.1216/RMJ-2008-38-4-1117.

Export citation


  • V.V. Chepyzhov and A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal. 4 (2005), 115-142.
  • V.V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptotic Anal. 46 (2006), 251-273.
  • M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal. 4 (2005), 705-720.
  • M. Conti, V. Pata and M. Squassina, Singular limit of dissipative hyperbolic equations with memory, Discrete Continuous Dynamical Systems, suppl. (2005), 200-208.
  • C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297-308.
  • --------, Contraction semigroups and trend to equilibrium in continuum mechanics, in Applications of methods of functional analysis to problems in mechanics, P. Germain and B. Nayroles, eds., Springer-Verlag, Berlin, 1976.%{pp.295--306, Lecture Notes in Mathematics no.503,
  • P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Continuous Dynamical Systems 10 (2004), 211-238.
  • M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal. 116 (1991), 139-152.
  • M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, SIAM Stud. Appl. Math. 12, SIAM, Philadelphia, {1992.
  • C. Giorgi, J.E. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl. 260 (2001), 83-99.
  • M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution equations, semi-groups and functional analysis, A. Lorenzi and B. Ruf, eds., Birkhäuser, Boston, 2002.
  • M.E. Gurtin and A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), 113-126.
  • J.K. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society, Providence, 1988.
  • O.A. Ladyzhenskaya, Finding minimal global attractors for the Navier-Stokes equations and other partial differential equations, Russian Math. Surveys 42 (1987), 27-73.
  • Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math. 54 (1996), 21-31.
  • --------, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes Math. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999.
  • V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math. 65 (2006), 499-513.
  • V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. 11 (2001), 505-529.
  • A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
  • M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical problems in viscoelasticity, Longman Scientific & Technical, Harlow John Wiley & Sons, Inc., New York, 1987.