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doi: 10.3934/mcrf.2022010
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Theoretical and computational decay results for a memory type wave equation with variable-exponent nonlinearity

1. 

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

2. 

Department of Mathematics, Research Groups MASEP and Bioinformatics FG, University of Sharjah, United Arab Emirates

*Corresponding author: Adel M. Al-Mahdi

Received  July 2021 Revised  January 2022 Early access March 2022

Fund Project: The first and second authors are supported by KFUPM grant SB201012

In this paper we are concerned with a viscoelastic wave equation with infinite memory and nonlinear frictional damping of variable-exponent type. First, we establish explicit and general decay results with a very general assumption on the relaxation function. Then, we remove the constraint imposed on the boundedness condition on the initial data used in the earlier results in the literature. Finally, we perform several numerical tests to illustrate our theoretical findings. This study generalizes and improves previous literature outcomes.

Citation: Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Mostafa Zahri. Theoretical and computational decay results for a memory type wave equation with variable-exponent nonlinearity. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022010
References:
[1]

M. AfilalA. Guesmia and M. Zahri, On the exponential and polynomial stability for a linear bresse system, Math. Methods Appl. Sci., 43 (2020), 2626-2645.  doi: 10.1002/mma.6070.

[2]

A. M. Al-Mahdi, Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity, Bound. Value Probl., 2020 (2020), 1-20.  doi: 10.1186/s13661-020-01382-9.

[3]

A. M. Al-Mahdi, General stability result for a viscoelastic plate equation with past history and general kernel, J. Math. Anal. Appl., 490 (2020), 124216, 19 pp. doi: 10.1016/j.jmaa.2020.124216.

[4]

A. M. Al-Mahdi and M. M. Al-Gharabli, New general decay results in an infinite memory viscoelastic problem with nonlinear damping, Bound. Value Probl., 2019 (2019), Paper No. 140, 15 pp. doi: 10.1186/s13661-019-1253-6.

[5]

A. M. Al-Mahdi, M. M. Al-Gharabli, A. Guesmia and S. A. Messaoudi, New decay results for a viscoelastic-type timoshenko system with infinite memory, Z. Angew. Math. Phys., 72 (2021), Paper No. 22, 24 pp. doi: 10.1007/s00033-020-01446-x.

[6]

A. M. Al-Mahdi, M. M. Al-Gharabli, M. Kafini and S. Al-Omari, On the global existence and asymptotic behavior of the solution of a nonlinear wave equation with past history, Journal of Mathematical Physics, 62 (2021), Paper No. 031512, 15 pp. doi: 10.1063/5.0003515.

[7]

F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems, J. Differential Equations, 248 (2010), 1473-1517.  doi: 10.1016/j.jde.2009.12.005.

[8]

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.  doi: 10.1007/s00245.

[9]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.

[10]

S. Antontsev and J. Ferreira, Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions, Nonlinear Anal., 93 (2013), 62-77.  doi: 10.1016/j.na.2013.07.019.

[11]

S. Antontsev and S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions, Atlantis Studies in Differential Equations, 4. Atlantis Press, Paris, 2015. doi: 10.2991/978-94-6239-112-3.

[12]

J. A. ApplebyM. FabrizioB. Lazzari and D. W. Reynolds, On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.  doi: 10.1142/S0218202506001674.

[13]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60. Springer-Verlag, New York-Heidelberg, 1978.

[14]

F. Baowei and Z. Mostafa, Optimal decay rate estimates of a nonlinear viscoelastic kirchhoff plate, Complexity, (2020), 6079507.

[15]

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

[16]

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.

[17]

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

[18]

L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[19]

M. FabrizioC. Giorgi and V. Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232.  doi: 10.1007/s00205-010-0300-3.

[20]

J. Ferreira and S. A. Messaoudi, On the general decay of a nonlinear viscoelastic plate equation with a strong damping and $\overrightarrow{p}(x, t)$-laplacian, Nonlinear Anal., 104 (2014), 40-49.  doi: 10.1016/j.na.2014.03.010.

[21]

C. GiorgiJ. E. M. 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.

[22]

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.

[23]

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.

[24]

A. Guesmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.

[25]

J. H. Hassan and S. A. Messaoudi, General decay results for a viscoelastic wave equation with a variable exponent nonlinearity, Asymptot. Anal., 125 (2021), 365-388.  doi: 10.3233/ASY-201661.

[26]

S. A. HassanJ. H. Messaoudi and M. Zahri, Existence and new general decay results for a viscoelastic-type timoshenko system, Z. Anal. Anwend., 39 (2020), 185-222.  doi: 10.4171/ZAA/1657.

[27]

K.-P. JinJ. Liang and T.-J. Xiao, Coupled second order evolution equations with fading memory: Optimal energy decay rate, J. Differ. Equ., 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018.

[28]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. 

[29]

W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. 

[30]

S. A. MessaoudiJ. H. Al-Smail and A. A. Talahmeh, Decay for solutions of a nonlinear damped wave equation with variable-exponent nonlinearities, Comput. Math. Appl., 76 (2018), 1863-1875.  doi: 10.1016/j.camwa.2018.07.035.

[31]

S. A. Messaoudi and A. A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Methods Appl. Sci., 40 (2017), 6976-6986.  doi: 10.1002/mma.4505.

[32]

S. A. MessaoudiA. A. Talahmeh and J. H. Al-Smail, Nonlinear damped wave equation: Exixstence and blow-up, Comput. Math. Appl., 74 (2017), 3024-3041.  doi: 10.1016/j.camwa.2017.07.048.

[33]

M. I. Mustafa, Energy decay in a quasilinear system with finite and infinite memories, Mathematical Methods in Engineering, 23 (2019), 235-256. 

[34]

M. I. Mustafa, Asymptotic stability for the second order evolution equation with memory, J. Dyn. Control Syst., 25 (2019), 263-273.  doi: 10.1007/s10883-018-9410-2.

[35]

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.

[36]

M. I. MustafaS. A. Messaoudi and M. Zahri, Theoretical and computational results of a wave equation with variable exponent and time-dependent nonlinear damping, Arab. J. Math., 10 (2021), 443-458.  doi: 10.1007/s40065-021-00312-6.

[37]

S.-H. Park and J.-R. Kang, Blow-up of solutions for a viscoelastic wave equation with variable exponents, Math. Meth. Appl. Sci., 42 (2019), 2083-2097.  doi: 10.1002/mma.5501.

[38]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.

[39]

M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[40]

V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Mehtods and Qualitative Analysis. Monographs And Research Notes In Mathematics, CRC Press, Boca Raton, FL, 2015. doi: 10.1201/b18601.

[41]

A. Youkana, Stability of an abstract system with infinite history, arXiv preprint, arXiv: 1805.07964, 2018.

show all references

References:
[1]

M. AfilalA. Guesmia and M. Zahri, On the exponential and polynomial stability for a linear bresse system, Math. Methods Appl. Sci., 43 (2020), 2626-2645.  doi: 10.1002/mma.6070.

[2]

A. M. Al-Mahdi, Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity, Bound. Value Probl., 2020 (2020), 1-20.  doi: 10.1186/s13661-020-01382-9.

[3]

A. M. Al-Mahdi, General stability result for a viscoelastic plate equation with past history and general kernel, J. Math. Anal. Appl., 490 (2020), 124216, 19 pp. doi: 10.1016/j.jmaa.2020.124216.

[4]

A. M. Al-Mahdi and M. M. Al-Gharabli, New general decay results in an infinite memory viscoelastic problem with nonlinear damping, Bound. Value Probl., 2019 (2019), Paper No. 140, 15 pp. doi: 10.1186/s13661-019-1253-6.

[5]

A. M. Al-Mahdi, M. M. Al-Gharabli, A. Guesmia and S. A. Messaoudi, New decay results for a viscoelastic-type timoshenko system with infinite memory, Z. Angew. Math. Phys., 72 (2021), Paper No. 22, 24 pp. doi: 10.1007/s00033-020-01446-x.

[6]

A. M. Al-Mahdi, M. M. Al-Gharabli, M. Kafini and S. Al-Omari, On the global existence and asymptotic behavior of the solution of a nonlinear wave equation with past history, Journal of Mathematical Physics, 62 (2021), Paper No. 031512, 15 pp. doi: 10.1063/5.0003515.

[7]

F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems, J. Differential Equations, 248 (2010), 1473-1517.  doi: 10.1016/j.jde.2009.12.005.

[8]

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.  doi: 10.1007/s00245.

[9]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.

[10]

S. Antontsev and J. Ferreira, Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions, Nonlinear Anal., 93 (2013), 62-77.  doi: 10.1016/j.na.2013.07.019.

[11]

S. Antontsev and S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions, Atlantis Studies in Differential Equations, 4. Atlantis Press, Paris, 2015. doi: 10.2991/978-94-6239-112-3.

[12]

J. A. ApplebyM. FabrizioB. Lazzari and D. W. Reynolds, On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.  doi: 10.1142/S0218202506001674.

[13]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60. Springer-Verlag, New York-Heidelberg, 1978.

[14]

F. Baowei and Z. Mostafa, Optimal decay rate estimates of a nonlinear viscoelastic kirchhoff plate, Complexity, (2020), 6079507.

[15]

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

[16]

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.

[17]

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

[18]

L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[19]

M. FabrizioC. Giorgi and V. Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232.  doi: 10.1007/s00205-010-0300-3.

[20]

J. Ferreira and S. A. Messaoudi, On the general decay of a nonlinear viscoelastic plate equation with a strong damping and $\overrightarrow{p}(x, t)$-laplacian, Nonlinear Anal., 104 (2014), 40-49.  doi: 10.1016/j.na.2014.03.010.

[21]

C. GiorgiJ. E. M. 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.

[22]

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.

[23]

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.

[24]

A. Guesmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.

[25]

J. H. Hassan and S. A. Messaoudi, General decay results for a viscoelastic wave equation with a variable exponent nonlinearity, Asymptot. Anal., 125 (2021), 365-388.  doi: 10.3233/ASY-201661.

[26]

S. A. HassanJ. H. Messaoudi and M. Zahri, Existence and new general decay results for a viscoelastic-type timoshenko system, Z. Anal. Anwend., 39 (2020), 185-222.  doi: 10.4171/ZAA/1657.

[27]

K.-P. JinJ. Liang and T.-J. Xiao, Coupled second order evolution equations with fading memory: Optimal energy decay rate, J. Differ. Equ., 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018.

[28]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. 

[29]

W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. 

[30]

S. A. MessaoudiJ. H. Al-Smail and A. A. Talahmeh, Decay for solutions of a nonlinear damped wave equation with variable-exponent nonlinearities, Comput. Math. Appl., 76 (2018), 1863-1875.  doi: 10.1016/j.camwa.2018.07.035.

[31]

S. A. Messaoudi and A. A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Methods Appl. Sci., 40 (2017), 6976-6986.  doi: 10.1002/mma.4505.

[32]

S. A. MessaoudiA. A. Talahmeh and J. H. Al-Smail, Nonlinear damped wave equation: Exixstence and blow-up, Comput. Math. Appl., 74 (2017), 3024-3041.  doi: 10.1016/j.camwa.2017.07.048.

[33]

M. I. Mustafa, Energy decay in a quasilinear system with finite and infinite memories, Mathematical Methods in Engineering, 23 (2019), 235-256. 

[34]

M. I. Mustafa, Asymptotic stability for the second order evolution equation with memory, J. Dyn. Control Syst., 25 (2019), 263-273.  doi: 10.1007/s10883-018-9410-2.

[35]

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.

[36]

M. I. MustafaS. A. Messaoudi and M. Zahri, Theoretical and computational results of a wave equation with variable exponent and time-dependent nonlinear damping, Arab. J. Math., 10 (2021), 443-458.  doi: 10.1007/s40065-021-00312-6.

[37]

S.-H. Park and J.-R. Kang, Blow-up of solutions for a viscoelastic wave equation with variable exponents, Math. Meth. Appl. Sci., 42 (2019), 2083-2097.  doi: 10.1002/mma.5501.

[38]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.

[39]

M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[40]

V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Mehtods and Qualitative Analysis. Monographs And Research Notes In Mathematics, CRC Press, Boca Raton, FL, 2015. doi: 10.1201/b18601.

[41]

A. Youkana, Stability of an abstract system with infinite history, arXiv preprint, arXiv: 1805.07964, 2018.

Figure 1.  The two dimensional wave behavior and the corresponding cut solutions (right)
Figure 2.  The two dimensional wave behavior (left) and the corresponding cut solutions (right)
Figure 3.  The energy functions
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