Decay result in a problem of a nonlinearly damped wave equation with variable exponent

Mohammad Kafini; Jamilu Hashim Hassan; Mohammad M. Al-Gharabli; Mohammad Kafini; Jamilu Hashim Hassan; Mohammad M. Al-Gharabli

doi:10.3934/math.2022170

AIMS Mathematics

2022, Volume 7, Issue 2: 3067-3082. doi: 10.3934/math.2022170

Previous Article Next Article

Research article

Decay result in a problem of a nonlinearly damped wave equation with variable exponent

1.
Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2.
The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3.
The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received: 19 August 2021 Accepted: 19 November 2021 Published: 24 November 2021
MSC : 35B37, 35L55, 74D05, 93D15, 93D20

In this work we study a wave equation with a nonlinear time dependent frictional damping of variable exponent type. The existence and uniqueness results are established using Fadeo-Galerkin approximation method. We also exploit the Komornik lemma to prove the uniform stability result for the energy associated to the solution of the problem under consideration.
- viscoelasticity,
- relaxation function,
- general decay,
- logarithmic nonlinearity,
- variable exponent
Citation: Mohammad Kafini, Jamilu Hashim Hassan, Mohammad M. Al-Gharabli. Decay result in a problem of a nonlinearly damped wave equation with variable exponent[J]. AIMS Mathematics, 2022, 7(2): 3067-3082. doi: 10.3934/math.2022170

Related Papers:

Abstract

In this work we study a wave equation with a nonlinear time dependent frictional damping of variable exponent type. The existence and uniqueness results are established using Fadeo-Galerkin approximation method. We also exploit the Komornik lemma to prove the uniform stability result for the energy associated to the solution of the problem under consideration.

References

[1]	S. Antontsev, S. Shmarev, Evolution PDEs with nonstandard growth conditions: Existence, uniqueness, localization, blow-up, Paris: Atlantis Press, 2015.
[2]	Y. M. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383–1406. doi: 10.1137/050624522. doi: 10.1137/050624522
[3]	S. Antontsev, V. Zhikov, Higher integrability for parabolic equations of $p(x, t)$-Laplacian type, Adv. Differ. Equ., 10 (2005), 1053–1080.
[4]	M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403–417. doi: 10.1007/BF01444231. doi: 10.1007/BF01444231
[5]	A. Benaissa, S. A. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation, Nonlinear Differ. Equ. Appl., 12 (2006), 391–399. doi: 10.1007/s00030-005-0008-5. doi: 10.1007/s00030-005-0008-5
[6]	S. A. Messaoudi, M. I. Mustafa, General energy decay rates for a weakly damped wave equation, Commun. Math. Anal., 9 (2010), 67–76.
[7]	S. Ghegal, I. Hamchi, S. A. Messaoudi, Global existence and stability of a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal., 99 (2020), 1333–1343. doi: 10.1080/00036811.2018.1530760. doi: 10.1080/00036811.2018.1530760
[8]	I. Lasiecka, Stabilization of wave and plate-like equation with nonlinear dissipation on the boundary, J. Differ. Equ., 79 (1989), 340–381. doi: 10.1016/0022-0396(89)90107-1. doi: 10.1016/0022-0396(89)90107-1
[9]	I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507–533.
[10]	M. Nakao, Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations, Math Z., 206 (1991), 265–275. doi: 10.1007/BF02571342. doi: 10.1007/BF02571342
[11]	M. Kafini, S. A. Messaoudi, On the decay and global nonexistence of solutions to a damped wave equation with variable-exponent nonlinearity and delay, Ann. Pol. Math., 122 (2019), 49–70.
[12]	S. A. Messaoudi, On the decay of solutions of a damped quasilinear wave equation with variable-exponent nonlinearities, Math. Meth. Appl. Sci., 43 (2020), 5114–5126. doi: 10.1002/mma.6254. doi: 10.1002/mma.6254
[13]	S. Antontsev, Wave equation with $p(x, t)$-Laplacian and damping term: Blow-up of solutions, C. R. Mecanique, 339 (2011), 751–755. doi: 10.1016/j.crme.2011.09.001. doi: 10.1016/j.crme.2011.09.001
[14]	S. Antontsev, J. Ferreira, Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions, Nonlinear Anal.-Theor., 93 (2013), 62–77. doi: 10.1016/j.na.2013.07.019. doi: 10.1016/j.na.2013.07.019
[15]	B. Guo, W. J. Gao, Blow-up of solutions to quasilinear hyperbolic equations with $p(x, t)$-Laplacian and positive initial energy, C. R. Mecanique, 342 (2014), 513–519. doi: 10.1016/j.crme.2014.06.001. doi: 10.1016/j.crme.2014.06.001
[16]	S. A. Messaoudi, A. A. Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal., 96 (2017), 1509–1515. doi: 10.1080/00036811.2016.1276170. doi: 10.1080/00036811.2016.1276170
[17]	S. A. Messaoudi, A. A. Talahmeh, On wave equation: Review and recent results, Arab. J. Math., 7 (2018), 113–145. doi: 10.1007/s40065-017-0190-4. doi: 10.1007/s40065-017-0190-4
[18]	H. G. Sun, A. L. Chang, Y. Zhang, W. Chen, A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal., 22 (2018), 27–59. doi: 10.1515/fca-2019-0003. doi: 10.1515/fca-2019-0003
[19]	X. C. Zheng, H. Wang, Analysis and discretization of a variable-order fractional wave equation, Commun. Nonlinear Sci., 104 (2022), 106047. doi: 10.1016/j.cnsns.2021.106047. doi: 10.1016/j.cnsns.2021.106047
[20]	X. C. Zheng, H. Wang, An error estimate of a numerical approximation to a Hidden-memory variable-order space-time fractional diffusion equation, SIAM J. Numer. Anal., 58 (2020), 2492–2514. doi: 10.1137/20M132420X. doi: 10.1137/20M132420X
[21]	X. C. Zheng, H. Wang, A Hidden-memory variable-order time-fractional optimal control model: Analysis and approximation, SIAM J. Control Optim., 59 (2021), 1851–1880. doi: 10.1137/20M1344962. doi: 10.1137/20M1344962
[22]	X. C. Zheng, H. Wang, Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal., 41 (2021), 1522–1545. doi: 10.1093/imanum/draa013. doi: 10.1093/imanum/draa013
[23]	L. Diening, P. Harjulehto, P. Hästö, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Berlin, Heidelberg: Springer-Verlag, 2011. doi: 10.1007/978-3-642-18363-8.
[24]	X. L. Fan, D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $ W^{m, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424–446. doi: 10.1006/jmaa.2000.7617. doi: 10.1006/jmaa.2000.7617
[25]	J. L. Lions, Quelques méthodes de résolution des problemes aux limites nonlinéaires, Paris: Dunod, 1969.
[26]	M. T. Lacroix-Sonrier, Distrubutions, espaces de sobolev: Applications, Paris: Ellipses, 1998.
[27]	V. Komornik, Exact controllability and stabilization. The multiplier method, Paris: Masson-John Wiley, 1994.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)