# American Institute of Mathematical Sciences

July  2014, 13(4): 1541-1551. doi: 10.3934/cpaa.2014.13.1541

## General decay estimates for a Cauchy viscoelastic wave problem

 1 Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900 2 King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261

Received  August 2013 Revised  December 2013 Published  February 2014

In this paper, we consider the Cauchy problem of a viscoelatic wave equation and by using the energy method in the Fourier space, we show general decay estimates of the solution. This result improves and generalizes some other results in the literature.
Citation: Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541
##### References:
 [1] S. Berrimi and S. A. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differential Equations, pages No. 88, 10 pp. (electronic), 2004. [2] S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonl. Anal., 64 (2006), 2314-2331. doi: 10.1016/j.na.2005.08.015. [3] E. L. Cabanillas and J. E. Mu noz Rivera, Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels, Comm. Math. Phys., 177 (1996), 583-602. [4] M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193. [5] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, E. J. Differential Equations, 44 (2002), 1-14. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma and J. A. Soriano, Global existence and asymptotic stability for viscoelastic problems, Differential Integral Equations, 15 (2002), 731-748. [7] M. Conti, S. Gatti and V. Pata, Decay rates of volterra equations on $\mathbbR^n$, Cent. Eur. J. Math., 5 (2007), 720-732. doi: 10.2478/s11533-007-0024-2. [8] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. [9] C. M. Dafermos, On abstract volterra equations with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569. [10] G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized $3$-d viscoelasticity, Quart. Appl. Math., 48 (1990), 715-730. [11] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 693-698. [12] X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Methods Appl. Sci., 32 (2009), 346-358. doi: 10.1002/mma.1041. [13] W. J. Hrusa, Global existence and asymptotic stability for a semilinear hyperbolic Volterra equation with large initial data, SIAM J. Math. Anal., 16 (1985), 110-134. doi: 10.1137/0516007. [14] R. Ikehata, Decay estimates by moments and masses of initial data for linear damped wave equations, Int. J. Pure Appl. Math., 5 (2003), 77-94. [15] W. Liu, General decay of solutions to a viscoelastic wave equation with nonlinear localized damping, Ann. Acad. Sci. Fenn. Math., 34 (2009), 291-302. [16] W. Liu, General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms, J. Math. Phys., 50 (2009), 113506. doi: 10.1063/1.3254323. [17] Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinetic. Related. Models., 4 (2011), 531-547. doi: 10.3934/krm.2011.4.531. [18] S. A. Messaoudi, General decay of solution energy in a viscoelastic equation with a nonlinear source, Nonl. Anal., 69 (2008), 2589-2598. doi: 10.1016/j.na.2007.08.035. [19] S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467. [20] S. A. Messaoudi and M. I. Mustafa, On the control of solutions of viscoelastic equations with boundary feedback, Nonlinear Anal. Real World Appl., 10 (2009), 3132-3140. doi: 10.1016/j.nonrwa.2008.10.026. [21] J. E. Mu noz Rivera and M. G. Naso, On the decay of the energy for systems with memory and indefinite dissipation, Asymptot. Anal., 49 (2006), 189-204. [22] J. Y. Park and S. H. Park, General decay for quasilinear viscoelastic equations with nonlinear weak damping, J. Math. Phys., 50 (2009), 083505. doi: 10.1063/1.3187780. [23] R. Racke and B. Said-Houari, Decay rates for semilinear viscoelastic systems in weighted spaces, J. Hyperbolic Differ. Equ., 9 (2012), 67-103. doi: 10.1142/S0219891612500026. [24] R. Racke and B. Said-Houari, Decay rates and global existence for semilinear dissipative Timoshenko systems, Quart. Appl. Math., 72 (2013), 229-266. doi: 10.1090/S0033-569X-2012-01280-8. [25] J. E. Mu noz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. [26] J. E. Mu noz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704. [27] J. E. Mu noz Rivera and A. Peres Salvatierra, Asymptotic behavior of the energy in partially viscoelastic materials, Quart. Appl. Math., 59 (2001), 557-578. [28] B. Said-Houari, Diffusion phenomenon for linear dissipative wave equations, Z. Anal. Anwend., 31 (2012), 267-282. doi: 10.4171/ZAA/1459. [29] J. Wirth, Asymptotic properties of solutions to wave equations with time-dependent dissipation, PhD thesis, TU Bergakademie Freiberg, 2004.

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##### References:
 [1] S. Berrimi and S. A. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differential Equations, pages No. 88, 10 pp. (electronic), 2004. [2] S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonl. Anal., 64 (2006), 2314-2331. doi: 10.1016/j.na.2005.08.015. [3] E. L. Cabanillas and J. E. Mu noz Rivera, Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels, Comm. Math. Phys., 177 (1996), 583-602. [4] M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193. [5] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, E. J. Differential Equations, 44 (2002), 1-14. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma and J. A. Soriano, Global existence and asymptotic stability for viscoelastic problems, Differential Integral Equations, 15 (2002), 731-748. [7] M. Conti, S. Gatti and V. Pata, Decay rates of volterra equations on $\mathbbR^n$, Cent. Eur. J. Math., 5 (2007), 720-732. doi: 10.2478/s11533-007-0024-2. [8] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. [9] C. M. Dafermos, On abstract volterra equations with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569. [10] G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized $3$-d viscoelasticity, Quart. Appl. Math., 48 (1990), 715-730. [11] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 693-698. [12] X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Methods Appl. Sci., 32 (2009), 346-358. doi: 10.1002/mma.1041. [13] W. J. Hrusa, Global existence and asymptotic stability for a semilinear hyperbolic Volterra equation with large initial data, SIAM J. Math. Anal., 16 (1985), 110-134. doi: 10.1137/0516007. [14] R. Ikehata, Decay estimates by moments and masses of initial data for linear damped wave equations, Int. J. Pure Appl. Math., 5 (2003), 77-94. [15] W. Liu, General decay of solutions to a viscoelastic wave equation with nonlinear localized damping, Ann. Acad. Sci. Fenn. Math., 34 (2009), 291-302. [16] W. Liu, General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms, J. Math. Phys., 50 (2009), 113506. doi: 10.1063/1.3254323. [17] Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinetic. Related. Models., 4 (2011), 531-547. doi: 10.3934/krm.2011.4.531. [18] S. A. Messaoudi, General decay of solution energy in a viscoelastic equation with a nonlinear source, Nonl. Anal., 69 (2008), 2589-2598. doi: 10.1016/j.na.2007.08.035. [19] S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467. [20] S. A. Messaoudi and M. I. Mustafa, On the control of solutions of viscoelastic equations with boundary feedback, Nonlinear Anal. Real World Appl., 10 (2009), 3132-3140. doi: 10.1016/j.nonrwa.2008.10.026. [21] J. E. Mu noz Rivera and M. G. Naso, On the decay of the energy for systems with memory and indefinite dissipation, Asymptot. Anal., 49 (2006), 189-204. [22] J. Y. Park and S. H. Park, General decay for quasilinear viscoelastic equations with nonlinear weak damping, J. Math. Phys., 50 (2009), 083505. doi: 10.1063/1.3187780. [23] R. Racke and B. Said-Houari, Decay rates for semilinear viscoelastic systems in weighted spaces, J. Hyperbolic Differ. Equ., 9 (2012), 67-103. doi: 10.1142/S0219891612500026. [24] R. Racke and B. Said-Houari, Decay rates and global existence for semilinear dissipative Timoshenko systems, Quart. Appl. Math., 72 (2013), 229-266. doi: 10.1090/S0033-569X-2012-01280-8. [25] J. E. Mu noz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. [26] J. E. Mu noz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704. [27] J. E. Mu noz Rivera and A. Peres Salvatierra, Asymptotic behavior of the energy in partially viscoelastic materials, Quart. Appl. Math., 59 (2001), 557-578. [28] B. Said-Houari, Diffusion phenomenon for linear dissipative wave equations, Z. Anal. Anwend., 31 (2012), 267-282. doi: 10.4171/ZAA/1459. [29] J. Wirth, Asymptotic properties of solutions to wave equations with time-dependent dissipation, PhD thesis, TU Bergakademie Freiberg, 2004.
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