Abstract
In this paper, we study a nonlinear viscoelastic equation with damping. Global existence of weak solutions and the uniform decay estimates of the energy have been established.
Similar content being viewed by others
References
Ball, J.M.: Initial-boundary value problem for an extensible beam. J. Math. Anal. Appl. 42, 61–90 (1973)
Ball, J.M.: Stability theory for an extensible beam. J. Differ. Equ. 14, 399–418 (1973)
Biller, P.: Remark on the decay for generalized damped string and beam equations. Nonlinear Anal. 10, 839–842 (1986)
Brito, E.H.: Decay estimates for generalized damped extensible string and beam equation. Nonlinear Anal. 8, 1489–1496 (1984)
Cavalcanti, M.M., Cavalcanti, V.D., Ferreira, J.: Existence and uniform decay for a non-linear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24, 1043–1053 (2001)
Cavalcanti, M.M., Cavalcanti, V.D., Soriano, J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron J. Differ. Equ. 2002, 1–14 (2002)
Han, X., Wang, M.: Global existence and uniform decay for a non-linear viscoelastic equation with damping. Nonlinear Anal. 70, 3090–3098 (2009)
Lagnese, J.: Asymptotic Energy Estimates for Kirchhoff Plates Subject to Weak Viscoelastic Damping. International Series of Numerical Mathematics, vol. 91. Birkhäuser, Basel (1989)
Lagnese, J.: Boundary Stabilization of Thin Plates. SIAM, Philadelphia (1989)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problem and Applications, vol. I. Springer, New York (1972)
Messaoudi, S.A., Tatar, N.: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods Appl. Sci. 30, 665–680 (2007)
Park, J.Y., Bae, J.J., Jung, I.H.: Uniform decay of solution for wave equation of Kirchhoff type with nonlinear boundary damping and memory term. Nonlinear Anal. 50, 871–884 (2002)
Santos, M.L., Junior, F.: A boundary condition with memory for Kirchhoff plates equations. Appl. Math. Comput. 148, 475–496 (2004)
Santos, M.L., Rerreira, J., Raposo, C.A.: Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary. Abstr. Appl. Anal. 2005(8), 901–919 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Park, J.Y., Kang, J.R. Global Existence and Uniform Decay for a Nonlinear Viscoelastic Equation with Damping. Acta Appl Math 110, 1393–1406 (2010). https://doi.org/10.1007/s10440-009-9516-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-009-9516-3