Research article

Lower bounds for the blow-up time to a nonlinear viscoelastic wave equation with strong damping

  • Received: 19 August 2018 Accepted: 11 October 2018 Published: 01 November 2018
  • This paper deals with a nonlinear viscoelastic wave equation with strong damping. By the means of the interpolation inequalities and di erential inequality technique, we obtain a lower bound for blow-up time of the solution. This result extends our earlier work Peng et al. [Appl. Math. Lett., 76, 2018].

    Citation: Xiaoming Peng, Xiaoxiao Zheng, Yadong Shang. Lower bounds for the blow-up time to a nonlinear viscoelastic wave equation with strong damping[J]. AIMS Mathematics, 2018, 3(4): 514-523. doi: 10.3934/Math.2018.4.514

    Related Papers:

  • This paper deals with a nonlinear viscoelastic wave equation with strong damping. By the means of the interpolation inequalities and di erential inequality technique, we obtain a lower bound for blow-up time of the solution. This result extends our earlier work Peng et al. [Appl. Math. Lett., 76, 2018].


    加载中
    [1] A. B. Al'shin, M. O. Korpusov, A. G. Siveshnikov, Blow up in nonlinear Sobolev type equations, Series in Nonlinear Analysis and Applications, Vol. 15, Berlin: De Gruyter, 2011.
    [2] S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58–66.
    [3] S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320 (2006), 902–915.
    [4] S. T. Wu, Blow-up of solutions for an integro-di_erential equation with a nonlinear source, Electron. J. Di_er. Eq., 45 (2006), 1–9.
    [5] H. T. Song, C. K. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Analysis: Real World Applications, 11 (2010), 3877–3883.
    [6] H. T. Song, D. X. Xue, Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Analysis: Theory, Methods & Applications, 109 (2014), 245–251.
    [7] F. S. Li, C. L. Zhao, Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3468–3477.
    [8] F. S. Li, Z. Q. Zhao, Y. F. Chen, Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, Nonlinear Analysis: Real World Applications, 12 (2011), 1759–1773.
    [9] W. Liu, Y. Sun, G. Li, On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term, Topol. Method Nonl. An., 49 (2017), 299–323.
    [10] W. Liu, D. Wang, D. Chen, General decay of solution for a transmission problem in infinite memory-type thermoelasticity with second sound, J. Therm. Stresses, 41 (2018), 758–775.
    [11] S. H. Park, M. J. Lee, J. R. Kang, Blow-up results for viscoelastic wave equations with weak damping, Appl. Math. Lett., 80 (2018), 20–26.
    [12] L. E. Payne, P.W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl., 328 (2007), 1196–1205.
    [13] L. E. Payne, P.W. Schaefer, Lower bounds for blow-up time in parabolic problems under Neumann conditions, J. Math. Appl. Anal., 85 (2006), 1301–1311.
    [14] L. L. Sun, B. Guo, W. J. Gao, A lower bound for the blow-up time to a damped semilinear wave equation, Appl. Math. Lett., 37 (2014), 22–25.
    [15] G. A. Philippin, Lower bounds for blow-up time in a class of nonlinear wave equations, Z. Angew. Math. Phys., 66 (2014), 129–134.
    [16] G. A. Philippin, S.Vernier Piro, Lower bound for the lifespan of solutions for a class of fourth order wave equations, Appl. Math. Lett., 50 (2015), 141–145.
    [17] B. Guo, F. Liu, A lower bound for the blow-up time to a viscoelastic hyperbolic equation with nonlinear sources, Appl. Math. Lett., 60 (2016), 115–119.
    [18] K. Baghaei, Lower bounds for the blow-up time in a superlinear hyperbolic equation with linear damping term, Comput. Math. Appl., 73 (2017), 560–564.
    [19] L. Yang, F. Liang, Z. H. Guo, Lower bounds for blow-up time of a nonlinear viscoelastic wave equation, Bound. Value Probl., 2015 (2015), 219.
    [20] S. Y. Tian, Bounds for blow-up time in a semilinear parabolic problem with viscoelastic term, Comput. Math. Appl., 74 (2017), 736–743.
    [21] X. M. Peng, Y. D. Shang, X. X. Zheng, Lower bounds for the blow-up time to a nonlinear viscoelastic wave equation with strong damping, Appl. Math. Lett., 76 (2018), 66–73.
    [22] R. A. Adams, J. J. F. Fournier, Sobolev Spaces, 2Eds., New York: Academic Press, 2003.
  • Reader Comments
  • © 2018 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(3390) PDF downloads(670) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog