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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

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