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January  2021, 20(1): 389-404. doi: 10.3934/cpaa.2020273

New general decay result for a system of viscoelastic wave equations with past history

Adel M. Al-Mahdi 1,, , Mohammad M. Al-Gharabli 1, and  Salim A. Messaoudi 2,

1. 

The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2. 

Department of Mathematics, University of Sharjah, P. O. Box, 27272, Sharjah. UAE

*Corresponding author

Received  November 2019 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: This work is funded by KFUPM under Project #SB191037

This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels
$ g_i : [0, +\infty) \rightarrow (0, +\infty) $
 satisfying
$ g_i'(t)\leq-\xi_i(t)H_i(g_i(t)),\qquad\forall\,t\geq0 \quad\mathrm{and\ for\ }i = 1,2, $
where
$ \xi_i $
 and
$ H_i $
 are functions satisfying some specific properties. Under this very general assumption on the behavior of
$ g_i $
 at infinity, we establish a relation between the decay rate of the solutions and the growth of
$ g_i $
 at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.
Keywords: Infinite memory, general decay, convex functions, viscoelastic equations, relaxation function.
Mathematics Subject Classification: Primary: 35B35, 35B40, 93D20.
Citation: Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure and Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273
References:
[1]

M. Al-Gharabli and M. Kafini, A general decay result of a coupled system of nonlinear wave equations, Rend. Circ. Mat. Palermo, II (2017), 1-13.  doi: 10.1007/s12215-017-0301-2.

[2]

A. Al-Mahdi and M. Al-Gharabli, New general decay results in an infinite memory viscoelastic problem with nonlinear damping, Bound. Value Probl., 2019 (2019), 140. doi: 10.1186/s13661-019-1253-6.

[3]

D. Andrade and A. Mognon, Global Solutions for a System of Klein- Gordon Equations with Memory, Bol. Soc. Paran. Mat, 21 (2003), 127-138.  doi: 10.5269/bspm.v21i1-2.7512.

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[5]

F. Belhannache, M. Algharabli and S. Messaoudi, Asymptotic stability for a viscoelastic equation with nonlinear damping and very general type of relaxation function, J. Dyn. Control Sys., 26 (2020), 45-67.  doi: 10.1007/s10883-019-9429-z.

[6]

S. Berrimi and S. A. Messaoudi, Exponential Decay of Solutions To a Viscoelastic. Electron, J. Differ. Equ., 2004 (2004), 1-10. 

[7]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.  doi: 10.3934/cpaa.2005.4.705.

[8]

C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differ. Equ., 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[10]

C. Giorgi, J. Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.  doi: 10.1006/jmaa.2001.7437.

[11]

A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760.  doi: 10.1016/j.jmaa.2011.04.079.

[12]

A. Guesmia, New general decay rates of solutions for two viscoelastic wave equations with infinite memory, Math. Model. Anal., 25 (2020), 351-373.  doi: 10.3846/mma.2020.10458.

[13]

A. Guesmia and N. Tatar, Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay, Hal-Inria, (2015). doi: 10.3934/cpaa.2015.14.457.

[14]

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.

[15]

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.

[16]

W. Liu, General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms, J. Math. Phys., 50 (2019), 113506. doi: 10.1063/1.3254323.

[17]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.  doi: 10.1016/j.jmaa.2007.11.048.

[18]

S. A. Messaoudi and M. M. Al-Gharabli, A general decay result of a nonlinear system of wave equations with infinite memories, Appl. Math. Comput., 259 (2015), 540-551.  doi: 10.1016/j.amc.2015.02.085.

[19]

S. A. Messaoudi and M. M. Al-Gharabli, A general stability result for a nonlinear wave equation with infinite memory, Appl. Math. Lett., 26 (2013), 1082-1086.  doi: 10.1016/j.aml.2013.06.002.

[20]

S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Appl. Math. Lett., 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.

[21]

S. A. Messaoudi and J. Hassan, On the general decay for a system of viscoelastic wave equations, In Current Trends in Mathematical Analysis and Its Interdisciplinary Applications, (2019), 287–310.

[22]

S. A. Messaoudi and N. Tatar, Uniform stabilization of solutions of a nonlinear system of viscoelastic equations, Appl. Anal., 87 (2008), 247-263.  doi: 10.1080/00036810701668394.

[23]

J. E. Rivera and E. C. Lapa, Decay rates of solutions of an anisotropic inhomogeneousn-dimensional viscoelastic equation with polynomially decaying kernels, Commun. Math. Phys., 177 (1996), 583-602. 

[24]

J. E. Munoz Rivera and J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Q. Appl. Math., 52 (1994), 629-648.  doi: 10.1090/qam/1306041.

[25]

M. I. Mustafa, Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations, Nonlinear Anal., 3 (2012), 452-463.  doi: 10.1016/j.nonrwa.2011.08.002.

[26]

M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.

[27]

B. Said-Houari, S. A. Messaoudi and A. Guesmia, General decay of solutions of a nonlinear system of viscoelastic wave equations, Nonlinear Differ. Equ. Appl., 18 (2011), 659-684.  doi: 10.1007/s00030-011-0112-7.

[28]

M. L. Santos, Decay rates for solutions of a system of wave equations with memory, Electron. J. Differ. Equ., 2002 (2002), 1-17. 

[29]

I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. Fr., 91 (1963), 129-135. 

show all references

References:
[1]

M. Al-Gharabli and M. Kafini, A general decay result of a coupled system of nonlinear wave equations, Rend. Circ. Mat. Palermo, II (2017), 1-13.  doi: 10.1007/s12215-017-0301-2.

[2]

A. Al-Mahdi and M. Al-Gharabli, New general decay results in an infinite memory viscoelastic problem with nonlinear damping, Bound. Value Probl., 2019 (2019), 140. doi: 10.1186/s13661-019-1253-6.

[3]

D. Andrade and A. Mognon, Global Solutions for a System of Klein- Gordon Equations with Memory, Bol. Soc. Paran. Mat, 21 (2003), 127-138.  doi: 10.5269/bspm.v21i1-2.7512.

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[5]

F. Belhannache, M. Algharabli and S. Messaoudi, Asymptotic stability for a viscoelastic equation with nonlinear damping and very general type of relaxation function, J. Dyn. Control Sys., 26 (2020), 45-67.  doi: 10.1007/s10883-019-9429-z.

[6]

S. Berrimi and S. A. Messaoudi, Exponential Decay of Solutions To a Viscoelastic. Electron, J. Differ. Equ., 2004 (2004), 1-10. 

[7]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.  doi: 10.3934/cpaa.2005.4.705.

[8]

C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differ. Equ., 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[10]

C. Giorgi, J. Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.  doi: 10.1006/jmaa.2001.7437.

[11]

A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760.  doi: 10.1016/j.jmaa.2011.04.079.

[12]

A. Guesmia, New general decay rates of solutions for two viscoelastic wave equations with infinite memory, Math. Model. Anal., 25 (2020), 351-373.  doi: 10.3846/mma.2020.10458.

[13]

A. Guesmia and N. Tatar, Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay, Hal-Inria, (2015). doi: 10.3934/cpaa.2015.14.457.

[14]

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.

[15]

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.

[16]

W. Liu, General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms, J. Math. Phys., 50 (2019), 113506. doi: 10.1063/1.3254323.

[17]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.  doi: 10.1016/j.jmaa.2007.11.048.

[18]

S. A. Messaoudi and M. M. Al-Gharabli, A general decay result of a nonlinear system of wave equations with infinite memories, Appl. Math. Comput., 259 (2015), 540-551.  doi: 10.1016/j.amc.2015.02.085.

[19]

S. A. Messaoudi and M. M. Al-Gharabli, A general stability result for a nonlinear wave equation with infinite memory, Appl. Math. Lett., 26 (2013), 1082-1086.  doi: 10.1016/j.aml.2013.06.002.

[20]

S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Appl. Math. Lett., 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.

[21]

S. A. Messaoudi and J. Hassan, On the general decay for a system of viscoelastic wave equations, In Current Trends in Mathematical Analysis and Its Interdisciplinary Applications, (2019), 287–310.

[22]

S. A. Messaoudi and N. Tatar, Uniform stabilization of solutions of a nonlinear system of viscoelastic equations, Appl. Anal., 87 (2008), 247-263.  doi: 10.1080/00036810701668394.

[23]

J. E. Rivera and E. C. Lapa, Decay rates of solutions of an anisotropic inhomogeneousn-dimensional viscoelastic equation with polynomially decaying kernels, Commun. Math. Phys., 177 (1996), 583-602. 

[24]

J. E. Munoz Rivera and J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Q. Appl. Math., 52 (1994), 629-648.  doi: 10.1090/qam/1306041.

[25]

M. I. Mustafa, Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations, Nonlinear Anal., 3 (2012), 452-463.  doi: 10.1016/j.nonrwa.2011.08.002.

[26]

M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.

[27]

B. Said-Houari, S. A. Messaoudi and A. Guesmia, General decay of solutions of a nonlinear system of viscoelastic wave equations, Nonlinear Differ. Equ. Appl., 18 (2011), 659-684.  doi: 10.1007/s00030-011-0112-7.

[28]

M. L. Santos, Decay rates for solutions of a system of wave equations with memory, Electron. J. Differ. Equ., 2002 (2002), 1-17. 

[29]

I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. Fr., 91 (1963), 129-135. 

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