Abstract
In this paper, we consider the nonlinear abstract equation
subject to a competing effect of viscoelastic and frictional dampings. With very general assumptions on the behavior of g at infinity and the behavior of h near 0, we establish explicit and optimal energy decay result. To the best of our knowledge, this is the first time we have such combination of generality and optimality in one explicit formula for the energy decay rates of this system.
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Alabau-Boussouira, F.: On convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51, 61–105 (2005)
Alabau-Boussouira, F., Cannarsa, P.: A general method for proving sharp energy decay rates for memory-dissipative evolution equations. C. R. Acad. Sci. Paris Ser. I 347, 867–872 (2009)
Alabau-Boussouira, F., Cannarsa, P., Sforza, D.: Decay estimates for the second order evaluation equation with memory. J. Funct. Anal. 245, 1342–1372 (2008)
Al-Gharabli, M.M., Al-Mahdi, A.M., Messaoudi, S.A.: General and optimal decay result for a viscoelastic problem with nonlinear boundary feedback. J. Dyn. Control Syst. 25(4), 551–572 (2019)
Al-Mahdi, A.M.: Optimal decay result for Kirchhoff plate equations with nonlinear damping and very general type of relaxation functions. Bound. Value Probl. 2019, 82 (2019). https://doi.org/10.1186/s13661-019-1196-y
Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer-Verlag, New York (1989)
Belhannache, F., Algharabli, M.M., Messaoudi, S.A.: Asymptotic stability for a viscoelastic equation with nonlinear damping and very general type of relaxation functions. J. Dyn. Control Syst. (2019). https://doi.org/10.1007/s10883-019-9429-z
Cabanillas, E.L., Munoz Rivera, J.E.: Decay rates of solutions of an anisotropic inhomogeneous \(n\)-dimensional viscoelastic equation with polynomial decaying kernels. Commun. Math. Phys. 177, 583–602 (1996)
Cannarsa, P., Sforza, D.: Integro-differential equations of hyperbolic type with positive definite kernels. J. Differ. Equ. 250, 4289–4335 (2011)
Cavalcanti, M.M., Cavalcanti, V.N.D., Lasiecka, I., Nascimento, F.A.: Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete Contin. Dyn. Syst. Ser. B 19(7), 1987–2012 (2014)
Cavalcanti, M.M., Cavalcanti, A.D.D., Lasiecka, I., Wang, X.: Existence and sharp decay rate estimates for a von Karman system with long memory. Nonlinear Anal. RWA 22, 289–306 (2015)
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.: On abstract Volterra equations with applications to linear viscoelasticity. J. Differ. Equ. 7, 554–569 (1970)
Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)
Fabrizio, M., Polidoro, S.: Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81(6), 1245–1264 (2002)
Guesmia, A.: Asymptotic stability of abstract dissipative systems with infinite memory. J. Math. Anal. Appl. 382, 748–760 (2011)
Guesmia, A., Messaoudi, S.A.: General energy decay estimates of Timoshenko system with frictional versus viscolastic damping. Math. Methods Appl. Sci. 32(16), 2102–2122 (2009)
Han, X., Wang, M.: General decay of energy for a viscoelastic equation with nonlinear damping. Math. Methods Appl. Sci. 32(3), 346–358 (2009)
Hrusa, W.J.: Global existence and asymptotic stability for a semilinear Volterra equation with large initial data. SIAM J. Math. Anal. 16(1), 110–134 (1985)
Jin, K.P., Liang, J., Xiao, T.J.: Coupled second order evolution equations with fading memory: optimal energy decay rate. J. Differ. Equ. 257, 1501–1528 (2014)
Kang, J.R.: Energy decay rates for von Karman system with memory and boundary feedback. Appl. Math. Comput. 218, 9085–9094 (2012)
Komornik, V.: Decay estimates for the wave equation with internal damping. Int. Ser. Numer. Math. 118, 253–266 (1994)
Lasiecka, I., Messaoudi, S.A., Mustafa, M.I.: Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys. 54, 031504 (2013). https://doi.org/10.1063/1.4793988
Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integral Equ. 8, 507–533 (1993)
Lasiecka, I., Wang, X.: Intrinsic decay rate estimates for semilinear abstract second order equations with memory. In: Favini, A., Fragnelli, G., Mininni, R. (eds.) New Prospects in Direct, Inverse and Control Problems for Evolution Equations. Springer INdAM Series, 271303, vol. 10. Springer, Cham (2014)
Liu, W.J.: General decay of solutions to a viscoelastic wave equation with nonlinear localized damping. Ann. Acad. Sci. Fenn. Math. 34(1), 291–302 (2009)
Liu, W.J.: General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J. Math. Phys. 50(11), art. No 113506 (2009)
Liu, W.-J., Zuazua, E.: Decay rates for dissipative wave equations. Ricerche Mat. 48, 61–75 (1999)
Martinez, P.: A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complut. 12(1), 251–283 (1999)
Messaoudi, S.A.: General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl. 341, 1457–1467 (2008)
Messaoudi, S.A., Mustafa, M.I.: On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Anal.: RWA 10, 3132–3140 (2009)
Munoz Rivera, J.E.: Asymptotic behavior in linear viscoelasticity. Q. Appl. Math. 52(4), 628–648 (1994)
Munoz Rivera, J.E., Naso, M.G., Vegni, F.M.: Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory. J. Math. Anal. Appl. 286(2), 692–704 (2003)
Mustafa, M.I.: General decay result for nonlinear viscoelastic equations. J. Math. Anal. Appl. 457, 134–152 (2018)
Mustafa, M.I.: On the control of the wave equation by memory-type boundary condition. Discrete Contin. Dyn. Syst. Ser. A 35(3), 1179–1192 (2015)
Mustafa, M.I.: Optimal decay rates for the viscoelastic wave equation. Math. Methods Appl. Sci. 41(1), 192–204 (2018)
Mustafa, M.I.: Uniform decay rates for viscoelastic dissipative systems. J. Dyn. Control Syst. 22(1), 101–116 (2016)
Mustafa, M.I.: Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations. Nonlinear Anal. RWA 13, 452–463 (2012)
Mustafa, M.I., Messaoudi, S.A.: General Energy decay rates for a weakly damped wave equation. Commun. Math. Anal. 9(2), 67–76 (2010)
Nakao, M.: Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305, 403–417 (1996)
Santos, M.L.: Asymptotic behavior of solutions to wave equations with a memory conditions at the boundary. Electron. J. Differ. Equ. 73, 1–11 (2001)
Xiao, T.J., Liang, J.: Coupled second order semilinear evolution equations indirectly damped via memory effects. J. Differ. Equ. 254(5), 2128–2157 (2013)
Zuazua, E.: Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equ. 15, 205–235 (1990)
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This work was supported by MASEP Research Group in the Research Institute of Sciences and Engineering at University of Sharjah.
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Mustafa, M.I. Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems. Z. Angew. Math. Phys. 72, 67 (2021). https://doi.org/10.1007/s00033-021-01498-7
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DOI: https://doi.org/10.1007/s00033-021-01498-7