Abstract
In this paper, we study the stabilization of a semi-linear viscoelastic wave equation subject to semi-linear and dynamical boundary conditions. The kernels used are of strongly positive definite type. We prove that internal and boundary memory damping are strong enough, via transmission process (\({u|_{\Gamma} = v}\)), to stabilize the whole system.
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Aassila M., Cavalcanti M.M., Soriano J.A.: Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a starshaped domain. SIAM J. Control Optim. 38(5), 1581–1602 (2000)
Alabau-Boussouira F., Cannarsa P., Sforza D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254, 1342–1372 (2008)
Cannarsa P., Sforza D.: integro-differential equations of hyperbolic type with positive definite kernels. J. Differ. Equ. 250, 4289–4335 (2011)
Cannarsa P., Sforza D.: A stability result for a class of nonlinear integrodifferential equations with L 1 kernels. Appl. Math. (Warsaw) 35, 395–430 (2008)
Cavalcanti M.M., Domingos Cavalcanti V.N., Ferreira J.: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24, 1043–1053 (2001)
Cavalcanti M.M., Domingos Cavalcanti V.N., Prates Filho J.S., Soriano J.A.: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integr. Equ. 14, 85–116 (2001)
Cavalcanti M.M., Domingos Cavalcanti V.N., 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., Khemmoudj A., Medjden M.: Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary condition. J. Math. Anal. Appl 328(2), 900–930 (2007)
Cavalcanti M.M., Oquendo H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)
Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Rational. Mech. Anal. 37, 297–308 (1970)
Dafermos C.M.: On abstract Volterra equation with applications to linear viscoeladticity. J. Differ. Equ. 7, 554–569 (1970)
Fabrizio M., Giorgi C., Pata V.: A new approach to equations with memory. Arch. Rational. Mech. Anal. 198, 189–232 (2010)
Gripenberg, G., Londen, S.O., Staffans, O.J.: Volterra Integral and Functional Equations. Encyclopedia Math. Appl., vol. 34. Cambridge Univ. Press, Cambridge (1990)
Hrusa W.J., Nohel J.A.: The Cauchy problem in one-dimensional nonlinear viscoelasticity. J. Differ. Equ. 59, 388–412 (1985)
Kawashima S.: Global solutions to the equation of viscoelasticity with fading memory. J. Differ. Equ. 101, 388–420 (1993)
Khemmoudj A., Medjden M.: Exponential decay for the semilinear damped Cauchy-Ventcel problem. Bol. Soc. Parana. Mat. 22(2), 97–116 (2004)
Lasiecka I., Triggiani R., Yao P.F.: Inverse/observability estimates for second-ordre hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235(1), 13–57 (1999)
Nicaise, S., Laoubi, K.: Polynomial stabilization of the wave equation with Ventcel’s boundary conditions. Math. Nachr. 283, 10 (2010)
Prüss, J.: Evolutionary Integral Equations and Applications. Monogr. Math., vol. 87. Birkhäuser Verlag, Basel (1993)
Staffans O.J.: Positive definite measures with applications to a Volterra equation. Trans. Am. Math. Soc. 218, 219–237 (1976)
Staffans O.J.: On a nonlinear hyperbolic Volterra equation. SIAM J. Math. Anal. 11, 793–812 (1980)
Zuazua E.: Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equ. 15(2), 205–235 (1990)
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Khemmoudj, A., Seghour, L. Exponential stabilization of a viscoelastic wave equation with dynamic boundary conditions. Nonlinear Differ. Equ. Appl. 22, 1259–1286 (2015). https://doi.org/10.1007/s00030-015-0322-5
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DOI: https://doi.org/10.1007/s00030-015-0322-5