Abstract
We consider the memory relaxation of an Euler–Bernoulli plate equation with nonlinear source term and internal frictional damping of arbitrary polynomial growth. The main focus is the existence of a smooth global attractor for the associated dynamical system.
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H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam 1973.
Cavalcanti M.M., Domingos Cavalcanti V.N., Ma T.F.: Exponential decay of the viscoelastic Euler–Bernoulli equation with a nonlocal dissipation in general domains. Diff. Int. Eqs. 17, 495–510 (2004)
V. V. Chepyzhov and M. I. Vishik, “Attractors for Equations of Mathematical Physics,” Amer. Math. Soc., Providence, 2002.
Chueshov D., Eller M., Lasiecka I.: On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Commun. in Partial Diff. Eqs. 27, 1901–1951 (2002)
I. Chueshov, I. Lasiecka, Long–time behavior of second–order evolutions with nonlinear damping Mem. Amer. Math. Soc. 195 (2008)
I. Chueshov, I. Lasiecka, Von Karman evolution equations: Well-posedness and long-time dynamics, Springer, New York, 2010.
Conti M., Gatti S., Pata V.: Uniform decay properties of linear Volterra integro-differential equations. Math. Models and Meth. Appl. Sci 18, 21–45 (2008)
Conti M., Marchini E.M.: Wave equations with memory: the minimal state approach, J. Math. Anal. Appl. 384, 607–625 (2011)
M. Conti, E.M. Marchini, V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst. 27 (2010), 1535–1552.
M. Conti, V. Pata, M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J. 55 (2006), 169–215.
C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal. 37 (1970), 554–569.
Fabrizio M., Giorgi C., Pata V.: A new approach to equations with memory. Arch. Ration. Mech. Anal. 198, 189–232 (2010)
P.G. Geredeli, I. Lasiecka, J.T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer, J. Diff. Eqs. 254 (2013), 1193–1229.
Geredeli P.G., Webster J.T.: Decay rates to equilibrium for nonlinear plate equations with degenerate, geometrically-constrained damping. Appl. Math. Optim. 68, 361–390 (2013)
Giorgi C., Pata V., Vuk E.: On the extensible viscoelastic beam. Nonlinearity 21, 713–733 (2008)
M. Grasselli, V. Pata, Uniform attractors of nonautonomous systems with memory, in “Evolution Equations, Semigroups and Functional Analysis” (A. Lorenzi and B. Ruf, Eds.), pp.155–178, Progr. Nonlinear Differential Equations Appl. no.50, Birkhäuser, Boston, 2002.
Greenberg J.M.: A priori estimates for flows in dissipative materials. J.Math.Anal. Appl. 60, 617–630 (1977)
J.K. Hale, Asymptotic behavior of dissipative systems, Amer. Math. Soc., Providence, 1988.
M.A. Horn, I. Lasiecka, Asymptotic behavior with respect to thichness of boundary stabilizing feedback for the Kirchoff plate, J. Diff. Eqs. 114 (1994), 396–433.
A.Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping, Nonlin. Anal. 72 (2010), 1993–1999.
J.E. Lagnese, “Boundary stabilization of thin plates”, Soc. for Industrial and Applied Math., Philadelphia, 1989.
Muoz Rivera J.E., Lapa E.C., Barreto R.: Decay rates for viscoelastic plates with memory. Journal of Elasticity 44, 61–87 (1996)
V. Narciso, Long-time behavior of a nonlinear viscoelastic beam equation with past history Math. Meth. Appl. Sci.(2014) in press.
Pata V.: Exponential stability in linear viscoelasticity. Quart. Appl. Math. 65, 499–513 (2006)
Pata V.: Stability and exponential stability in linear viscoelasticity. Milan J. Math. 77, 333–360 (2009)
Pata V.: Exponential stability in linear viscoelasticity with almost flat memory kernels. Comm. on Pure and Appl. Anal. 9, 721–730 (2010)
V. Pata, 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.
R. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, AMS, Providence, 1997.
R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer, New York, 1997.
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Pelin G. Geredeli was partially supported by the Scientific Research Projects Coordination Unit of Hacettepe University, Ankara.
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Conti, M., Geredeli, P.G. Existence of smooth global attractors for nonlinear viscoelastic equations with memory. J. Evol. Equ. 15, 533–558 (2015). https://doi.org/10.1007/s00028-014-0270-2
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DOI: https://doi.org/10.1007/s00028-014-0270-2