Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities

Mohammad M. Al-Gharabli; Adel M. Al-Mahdi; Mohammad Kafini; Mohammad M. Al-Gharabli; Adel M. Al-Mahdi; Mohammad Kafini

doi:10.3934/math.2021587

AIMS Mathematics

2021, Volume 6, Issue 9: 10105-10129. doi: 10.3934/math.2021587

Previous Article Next Article

Research article

Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities

1.
The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2.
Department of Mathematics and Statistics, 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: 12 May 2021 Accepted: 02 July 2021 Published: 09 July 2021
MSC : 35B37; 35L55; 74D05; 93D15; 93D20

In this paper, we consider the following viscoelastic problem with variable exponent and logarithmic nonlinearities:

$ u_{tt}-\Delta u+u+ \int_0^tb(t-s)\Delta u(s)ds+|u_t|^{{\gamma}(\cdot)-2}u_t = u\ln{\vert u\vert^{\alpha}}, $

where $ {\gamma}(.) $ is a function satisfying some conditions. We first prove a global existence result using the well-depth method and then establish explicit and general decay results under a wide class of relaxation functions and some specific conditions on the variable exponent function. Our results extend and generalize many earlier results in the literature.
- viscoelasticity,
- relaxation function,
- general decay,
- logarithmic nonlinearity,
- variable exponent
Citation: Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Mohammad Kafini. Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities[J]. AIMS Mathematics, 2021, 6(9): 10105-10129. doi: 10.3934/math.2021587

Related Papers:

Abstract

In this paper, we consider the following viscoelastic problem with variable exponent and logarithmic nonlinearities:

$ u_{tt}-\Delta u+u+ \int_0^tb(t-s)\Delta u(s)ds+|u_t|^{{\gamma}(\cdot)-2}u_t = u\ln{\vert u\vert^{\alpha}}, $

where $ {\gamma}(.) $ is a function satisfying some conditions. We first prove a global existence result using the well-depth method and then establish explicit and general decay results under a wide class of relaxation functions and some specific conditions on the variable exponent function. Our results extend and generalize many earlier results in the literature.

References

[1]	R. Christensen, Theory of viscoelasticity: An introduction, Elsevier, 1982.
[2]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differ. Eq., 44 (2002), 1-14.
[3]	S. A. Messaoudi, General decay of solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal. Theor., 69 (2008), 2589-2598. doi: 10.1016/j.na.2007.08.035
[4]	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
[5]	F. Alabau-Boussouira, P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad Sci. Paris, Ser I, 347 (2009), 867-872. doi: 10.1016/j.crma.2009.05.011
[6]	S. A. Messaoudi, 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
[7]	M. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Method. Appl. Sci., 41 (2017), 192-204.
[8]	S. Antontsev, S. Shmarev, Evolution PDEs with nonstandard growth conditions: Existence, uniqueness, localization, blow-up, Springer, 2015.
[9]	L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Springer, 2011.
[10]	S. Antontsev, Wave equation with $p(x, t)$-Laplacian and damping term: Existence and blow-up, Differ. Equ. Appl., 3 (2011), 503-525.
[11]	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
[12]	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
[13]	B. Guo, W. 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
[14]	S. A. Messaoudi, J. Al-Smail, 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
[15]	S. A. Messaoudi, A. Talahmeh, On wave equation: Review and recent results, Arab. J. Math., 7 (2018), 113-145. doi: 10.1007/s40065-017-0190-4
[16]	A. Palmieri, H. Takamura, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation II, Math. Nachr., 291 (2018), 1859-1892. doi: 10.1002/mana.201700144
[17]	A. Palmieri, M. Reissig, Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities, Nonlinear Anal., 187 (2019), 467-492. doi: 10.1016/j.na.2019.06.016
[18]	W. Chen, R. Ikehata, The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case, J. Differ. Equations, 292 (2021), 176-219. doi: 10.1016/j.jde.2021.05.011
[19]	J. Barrow, P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576-5587. doi: 10.1103/PhysRevD.52.5576
[20]	K. Enqvist, J. McDonald, Q-balls and baryogenesis in the MSSM, Phys. Lett. B, 425 (1998), 309-321. doi: 10.1016/S0370-2693(98)00271-8
[21]	K. Bartkowski, P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A Math. Theor., 41 (2008), 355201. doi: 10.1088/1751-8113/41/35/355201
[22]	I. Bialynicki-Birula, J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Pol. Sc., 23 (1975), 461-466.
[23]	P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Pol. B, 40 (2009), 59-66.
[24]	D. Edmunds, J. Rakosnik, Sobolev embeddings with variable exponent, Stud. Math., 143 (2000), 267-293. doi: 10.4064/sm-143-3-267-293
[25]	D. Edmunds, J. Rakosnik, Sobolev embeddings with variable exponent II, Math. Nachr., 246 (2002), 53-67.
[26]	X. 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
[27]	M. Al-Gharabli, A. Guesmia, S. A. Messaoudi, Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity, Appl. Anal., 99 (2020), 50-74. doi: 10.1080/00036811.2018.1484910
[28]	L. Gross, Logarithmic Sobolev inequalities, Am. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688
[29]	H. Chen, P. Luo, G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98. doi: 10.1016/j.jmaa.2014.08.030
[30]	F. Belhannache, M. Algharabli, 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
[31]	M. Al-Gharabli, A. Guesmia, Messaoudi S. A. Messaoudi, Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity, Commun. Pure Appl. Anal., 18 (2019), 159-180. doi: 10.3934/cpaa.2019009
[32]	V. Arnold, Mathematical methods of classical mechanics, Springer Science & Business Media, 1989.
[33]	J. Hassan, S. A. Messaoudi, General decay results for a viscoelastic wave equation with a variable exponent nonlinearity, Asymptotic Anal., 2021, 1-24.

Reader Comments

Your name:*

Email:*
© 2021 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)