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On optimal autocorrelation inequalities on the real line
New general decay result for a system of viscoelastic wave equations with past history
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 |
$ g_i : [0, +\infty) \rightarrow (0, +\infty) $ |
$ g_i'(t)\leq-\xi_i(t)H_i(g_i(t)),\qquad\forall\,t\geq0 \quad\mathrm{and\ for\ }i = 1,2, $ |
$ \xi_i $ |
$ H_i $ |
$ g_i $ |
$ g_i $ |
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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