Abstract
In this paper, we consider a nonlinear system of thermoviscoelastic with a nonlinear boundary damping and nonlinear source, in a bounded domain Ω, under appropriate assumptions imposed on relaxation function and with certain initial data, we establish the general decay rate of the solution energy, from which the usual exponential and polynomial decay are only special cases. This work generalizes and improves earlier results in the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aassila M., Cavalcanti M.M., Domingos Cavalcanti V.N.: Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term. Calc. Var. Partial Differ. Equ. 15(2), 155–180 (2002)
Alabau-Boussouira F.: Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51(1), 61–105 (2005)
Arnold V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)
Berrimi S., Messaoudi S.A.: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differ. Equ. 88, 1–10 (2004)
Cavalcanti M.M., Cavalcanti V.N.D., Prates Filho J.S., Soriano J.A.: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integral Equ. 14(1), 85–116 (2001)
Cavalcanti M.M., Cavalcanti V.N.D., Soriano J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 44, 1–14 (2002)
Cavalcanti M.M., Cavalcanti V.N.D., Martinez P.: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. Theory Methods Appl. 68(1), 177–193 (2008)
Li S., Zhao C.: Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping. Nonlinear Anal. Theory Methods Appl. 74(11), 3468–3477 (2011)
Liu W.J., Zuazua E.: Decay rates for dissipative wave equations. Ric. Math. 48, 61–75 (1999)
Lu L., Li S., Chai S.: On a viscoelastic equation with nonlinear boundary damping and source terms: global existence and decay of the solution. Nonlinear Anal. Real World Appl. 12(1), 295–303 (2011)
Messaoudi S.A., Mustafa M.I.: On convexity for energy decay rates of a viscoelastic equation with boundary feedback. Nonlinear Anal. Theory Methods Appl. 72(9-10), 3602–3611 (2010)
Park J.Y., Park S.H.: Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation. Czechoslov. Math. J. 56(2), 273–286 (2006)
Wu, S.-T., Chen, H.-F.: Uniform decay of solutions for a nonlinear viscoelastic wave equation with boundary dissipation. J. Funct. Spaces Appl. 2012:421847 (2012). doi:10.1155/2012/421847
Zhao F., Li Z., Chen Y.: Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation. Nonlinear Anal. Real World Appl. 12(3), 1759–1773 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boudiaf, A., Drabla, S. & Boulanouar, F. General Decay Rate for Nonlinear Thermoviscoelastic System with a Weak Damping and Nonlinear Source Term. Mediterr. J. Math. 13, 3101–3120 (2016). https://doi.org/10.1007/s00009-015-0674-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-015-0674-4