Research article

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

  • 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.

    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:

  • 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.



    加载中


    [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
  • © 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2547) PDF downloads(189) Cited by(12)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog