Abstract
We consider the viscoelastic plate equation and we prove that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially. When the kernel decays polynomially then the energy also decays polynomially. More precisely if the kernel g satisfies
g(t) ⩽ -c0g(t)1+1/p and g,g1+1/p ∈ L1(ℝ) with p > 2, then the energy decays as 1/(1+t)p.
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References
R.C. MacCamy, A model for one dimensional non linear viscoelasticity. Quart. Appl. Math. 35 (1977) 21–33.
C.M. Dafermos, An abstract Volterra equation with application to linear viscoelasticity. J. Diff. Eqs. 7 (1970) 554–589.
C.M. Dafermos, Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal., 37 (1970) 297–308.
C.M. Dafermos and J.A. Nohel, Energy methods for non linear hyperbolic Volterra integrodifferential equation. Comm. PDE 4 (1979) 219–278.
C.M. Dafermos and J.A. Nohel, A nonlinear hyperbolic Volterra equation in viscoelasticity. Amer. J. Math. Supplement (1981) 87–116.
G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized 3-D viscoelasticity. Quart. Appl. Math. 48 (1990) 715–730.
J.M. Greenberg, A priori estimates for flows in dissipative materials. J. Math. Anal. Appl. 60 (1977) 617–630.
W.J. Hrusa, Global existence and asymptotic stability for a semilinear Volterra equation with large initial data. SIAM J. Math. Anal. 16 (1), January 1985.
W.J. Hrusa and J.A. Nohel, The Cauchy problem in one dimensional nonlinear viscoelasticity. J. Diff. Eqs. 58 (1985) 388–412.
M. Renardy, W.J. Hursa and J.A. Nohel, Mathematical Problems in Viscoelasticity. Pitman monograph in Pure and Applied Mathematics 35 (1987).
J.E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping. International Series of Numerical Mathematics, Vol. 91, Birhäuser-Verlag, Bassel (1989).
J.E. Lagnese, and J.L. Lions, Modelling analysis of thin plates. Collection Recherches en Mathématiques Appliquées 6. Masson, Paris (1989).
J.L. Lions, and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, New York, Vol. I (1972).
J.E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. III (1994) 257–273.
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IM, Federal University of Rio de Janeiro, Supported by a grant of CNPq.
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Rivera, J.E.M., Lapa, E.C. & Barreto, R. Decay rates for viscoelastic plates with memory. J Elasticity 44, 61–87 (1996). https://doi.org/10.1007/BF00042192
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DOI: https://doi.org/10.1007/BF00042192