Abstract
In this paper, we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary and in the presence of infinite memory term. In the domain as well as on a part of the boundary, we use the multiplier method and some properties of the convex functions to prove an explicit and general decay result.
Similar content being viewed by others
References
Appleby J., Fabrizio M., Lazzari B., Reynolds D.: On exponential asymptotic stability in linear viscoelasticity. Math. Models Methods Appl. Sci. 16(10), 1677–1694 (2006)
Arnold V.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)
Berrimi, S., Messaoudi, S.: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differ. Equ. 88, 1–10 (2004)
Cavalcanti M., Domingos Cavalcanti V., Martinez P.: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 68(1), 177–193 (2008)
Cavalcanti M., Domingos Cavalcanti V., Prates Filho J., Soriano J.: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integral Equ. 14(1), 85–116 (2001)
Cavalcanti M., Domingos Cavalcanti V., Soriano J.A: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 44, 1–14 (2002)
Conti M., Pata V.: Weakly dissipative semilinear equations of viscoelasticity. Commun. Pure Appl. Anal. 4(4), 705–720 (2005)
Daoulatli M., Lasiecka I., Toundykov D.: Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. AIMS J. 2(1), 67–94 (2009)
Dasios G., Zafiropoulos F.: Equipartition of energy in linearized 3-D viscoelasticity. Quart. Appl. Math. 48(4), 715–730 (1990)
Fabrizio, M., Morro, A.: Mathematical Problems in Linear Viscoelasticity, vol. 12. Philadelphia: SIAM Studies in Applied Mathematics (1992)
Fabrizio M., Polidoro S.: Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81(6), 1245–1264 (2002)
Giorgi C., Muñoz Rivera J., Pata V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260(1), 83–99 (2001)
Guesmia A.: Asymptotic stability of abstract dissipative systems with infinite memory. J. Math. Appl. 382, 748–760 (2011)
Guesmia, A., Messaoudi, S.A.: A general decay result for a viscoelastic equation in the presence of past and finite history memories. Nonlinear Anal. Real World Appl. 13, 476–485 (2012)
Komornik V.: Exact Controllability and Stabilization. The Multiplier Method. Masson–Wiley, Paris (1994)
Lasiecka I., Tataru D.: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6(3), 507–533 (1993)
Messaoudi, S., Al-Gharabli, M.: A general stability result for a nonlinear wave equation with infinite memory. Appl. Math. Lett. 26, 1082–1086 (2013)
Messaoudi, S., Mustafa, M.: On convexity for energy decay rates of a viscoelastic equation with boundary feedback. Nonlinear Anal. 72, 3602–3611 (2010)
Muñoz Rivera J., Andrade D.: Exponential decay of nonlinear wave equation with a viscoelastic boundary condition. Math. Methods Appl. Sci. 23, 41–61 (2000)
Pata V.: Stability and exponential stability in linear viscoelasticity. Milan J. Math. 77(1), 333–360 (2009)
Santos, M.: Asymptotic behavior of solutions to wave equations with a memory condition at the boundary. Electron. J. Differ. Equ. 73, 1–11 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Messaoudi, S.A., Al-Gharabli, M.M. A general decay result of a viscoelastic equation with past history and boundary feedback. Z. Angew. Math. Phys. 66, 1519–1528 (2015). https://doi.org/10.1007/s00033-014-0476-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-014-0476-8