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Articles

Asymptotic behavior for coupled abstract evolution equations with one infinite memory

Pages 184-217 | Received 19 Aug 2013, Accepted 30 Jan 2014, Published online: 12 Mar 2014

References

  • Fabrizio M, Giorgi C, Pata V. A new approach to equations with memory. Arch. Rat. Mech. Anal. 2010;198:189–232.
  • Giorgi C, Muñoz Rivera JE, Pata V. Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 2001;260:83–99.
  • Muñoz Rivera JE, Naso MG. Optimal energy decay rate for a class of weakly dissipative second-order systems with memory. Appl. Math. Lett. 2010;23:743–746.
  • Pata V. Exponential stability in linear viscoelasticity. Quart. Appl. Math. 2006;3:499–513.
  • Chepyzhov VV, Pata V. Some remarks on stability of semigroups arising from linear viscoelasticity. Asymptot. Anal. 2006;46:251–273.
  • Dafermos CM. Asymptotic stability in viscoelasticity. Arch. Rat. Mech. Anal. 1970;37:297–308.
  • Fabrizio M, Lazzari B. On the existence and asymptotic stability of solutions for linear viscoelastic solids. Arch. Rat. Mech. Anal. 1991;116:139–152.
  • Guesmia A. Asymptotic stability of abstract dissipative systems with infinite memory. J. Math. Anal. Appl. 2011;382:748–760.
  • Liu Z, Zheng S. On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quart. Appl. Math. 1996;54:21–31.
  • Muñoz JE. Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory. J. Math Anal. Appl. 2007;326:691–707.
  • Pata V. Exponential stability in linear viscoelasticity with almost flat memory kernels. Commun. Pure. Appl. Anal. 2010;9:721–730.
  • Guesmia A, Messaoudi SA. A general decay result for a viscoelastic equation in the presence of past and finite history memories. Nonlinear Anal. 2012;13:476–485.
  • Guesmia A, Messaoudi SA, Soufyane A. On the stabilization for a linear Timoshenko system with infinite history and applications to coupled Timoshenko-heat systems. Elec. J. Diff. Equa. 2012;2012:1–45.
  • Messaoudi SA. General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl. 2008;341:1457–1467.
  • Messaoudi SA. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal. 2008;69:2589–2598.
  • Guesmia A, Messaoudi SA. General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math. Meth. Appl. Sci. 2009;32:2101–2122.
  • Guesmia A, Messaoudi SA. A general stability result in a Timoshenko system with infinite memory: A new approach. Math. Meth. Appl. Sci. 2014;37:384–392.
  • Guesmia A, Messaoudi SA, Said-Houari B. General decay of solutions of a nonlinear system of viscoelastic wave equations. NoDEA. 2011;18:659–684.
  • Guesmia A, Messaoudi SA. A new approach to the stability of an abstract hyperbolic system in the presence of infinite history. J. Math. Anal. Appl. Forthcoming. doi:10.1016/j.jmaa.2014.02.030
  • Cavalcanti MM, Domingos VN, Soriano JA. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Elec. J. Diff. Equa. 2002;44:1–14.
  • Muñoz Rivera JE, Fernández Sare HD. Stability of Timoshenko systems with past history. J. Math. Anal. Appl. 2008;339:482–502.
  • Berrimi S, Messaoudi SA. Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 2006;64:2314–2331.
  • Cavalcanti MM, Oquendo HP. Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 2003;42:1310–1324.
  • Messaoudi SA, Said-Houari B. Uniform decay in a Timoshenko-type with past history. J. Math. Anal. Appl. 2009;360:459–475.
  • Guesmia A, Messaoudi SA. On the stabilization of Timoshenko systems with memory and different speeds of wave propagation. Appl. Math. Comput. 2013;219:9424–9437.
  • Lasiecka I, Messaoudi SA, Mustafa MI. Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys. 2013;54:1–18.
  • Messaoudi SA, Mustafa MI. General stability result for viscoelastic wave equations. J. Math. Physics. 2012;53:1–14.
  • Tatar NE. Exponential decay for a viscoelastic problem with a singular kernel. Z. angew. Math. Phys. 2009;60:640–650.
  • Tatar NE. On a large class of kernels yielding exponential stability in viscoelasticity. Appl. Math. Comp. 2009;215:2298–2306.
  • Tatar NE. How far can relaxation functions be increasing in viscoelastic problems? Appl. Math. Lett. 2009;22:336–340.
  • Tatar NE. On a perturbed kernel in viscoelasticity. Appl. Math. Lett. 2011;24:766–770.
  • Tatar NE. Arbitrary decays in linear viscoelasticity. J. Math. Phys. 2011;52:1–12.
  • Tatar NE. A new class of kernels leading to an arbitrary decay in viscoelasticity. Mediterr. J. Math. 2010;6:139–150.
  • Tatar NE. Uniform decay in viscoelasticity for kernels with small non-decreasingness zones. Appl. Math. Comp. 2012;218:7939–7946.
  • Tatar NE. Oscillating kernels and arbitrary decays in viscoelasticity. Math. Nachr. 2012;285:1130–1143.
  • Russel DL. A general framework for the study of indirect damping mechanisms in elastic systems. J. Math. Anal. Appl. 1993;173:339–358.
  • Kapitonov BV. Uniform stabilization and exact controllability for a class of coupled hyperbolic systems. Comp. Appl. Math. 1996;15:199–212.
  • Alabau-Boussouira F, Cannarsa P, Komornik V. Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equa. 2002;2:127–150.
  • Guesmia A. Inégalités intégrales et applications à la stabilisation des systèmes distribués non dissipatifs [Integral inequalities and applications to the stabilization of nondissipative distributed systems] [Habilitation thesis]. France: University of Lorraine; 2006.
  • Guesmia A. Quelques résultats de stabilisation indirecte des systèmes couplés non dissipatifs [Some indirect stability results of nondissipative coupled systems]. Bull. Belg. Math. Soc. 2008;15:479–497.
  • Tebou L. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. MCRF. 2012;2:45–60.
  • Almeida RGC, Santos ML. Lack of exponential decay of a coupled system of wave equations with memory. Nonlinear Anal. 2011;12:1023–1032.
  • Komornik V. Exact controllability and stabilization. The multiplier method. Paris: Masson-John Wiley; 1994.
  • Liu Z, Zheng S. Semigroups associated with dissipative systems. Boca Raton, FL: Chapman Hall/CRC; 1999.
  • Pazy A. Semigroups of linear operators and applications to partial differential equations. New York (NY): Springer-Verlag; 1983.
  • Guesmia A. Contributions à la contrôlabilité exacte et la stabilisation des systèmes d’évolution [Contributions to the exact controllability and stabilization of evolutionary systems] [PhD Thesis]. France: Louis Pasteur University; 2000.

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