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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Blow up for a Nonlinear Viscoelastic Wave Equation with Strong Damping, Source and Delay Termes

Discontinuity, Nonlinearity, and Complexity 11(1) (2022) 139--148 | DOI:10.5890/DNC.2022.03.012

Abdelbaki Choucha , Djemal Ouchenane

Department of Mathematics, Faculty of Exact Sciences, University of El Oued, Algeria, Laboratory of pure and applied Mathematics, Amar Teledji Laghouat University, Algeria

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Abstract

In this work, we are concerned with a problem for a nonlinear viscoelastic wave equation with strong damping, source and delay terms, we proved a blow up result for the solution with negative initial energy under suitable conditions.

References

  1. [1]  Ball, J. (1977), Remarks on blow-up and nonexistence theorems for nonlinear evolutions equation, Quarterly Journal of Mathematics, 28, 473-486.
  2. [2] Berrimi, S. and Messaoudi, S. (2006), Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Analysis, 64, 2314-2331.
  3. [3]  Bialynicki-Birula, I. and Mycielsk, J. (1975), Wave equations with logarithmic nonlinearities, Bull Acad Polon Sci Ser SciMath Astron Phys, 23, 461-466.
  4. [4]  Boulaaras, S., Choucha, A., Ouchenane, D., and Cherif, B. (2020), Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms, Advances in Difference Equations, 2020, 310. https://doi.org/10.1186/s13662-020-02772-0.
  5. [5]  Cavalcanti, M.M., Cavalcanti, D., and Ferreira, J. (2001), Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24, 1043-1053.
  6. [6]  Cavalcanti, M.M., Cavalcanti, D., Filho, P.J.S., and Soriano, J.A. (2001), Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential and Integral Equations, 14, 85-116.
  7. [7] Cazenave, T. and Haraux, A. (1980), Equations d'evolution avec non-linearite logarithmique, Ann Fac Sci Toulousen Math (5); 2(1), 21-51.
  8. [8]  Choucha, A., Ouchenane, D., and Boulaaras, S. (2020), Blow-up of a nonlinear viscoelastic wave equation with distributed delay combined with strong damping and source terms, J. Nonlinear Funct. Anal., Article ID 31 https://doi.org/10.23952/jnfa.2020.31.
  9. [9]  Kafini, M. and Messaoudi, S.A. (2008), A blow-up result in a cauchy viscoelastic problem, Applied Mathematics Letters, 21, 549-553. http://dx.doi.org/10.1016/j.aml.2007.07.004.
  10. [10]  Kafini, M. and Messaoudi, S.A. (2018), Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay, Applicable Analysis, DOI: 10.1080/00036811.2018.1504029.
  11. [11] Guo, L., Yuan, Z., and Lin, G. (2015), Blow Up and Global Existence for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Nonlinear Damping and Source terms, Applied Mathematics, 6, 806-816.
  12. [12]  Messaoudi, S.A. (2006), Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, Journal of Mathematical Analysis and Applications, 320,902-915.
  13. [13] Nicaise, S. and Pignotti, C. (2008), Stabilization of the wave equation with boundary or internal distributed delay, Diff. Int. Equs., 21(9-10), 935-958.
  14. [14]  Piskin, E. and Yuksekkaya, H. (2020), Local existence and blow up of solutions for a logarithmic nonlinear viscoelastic wave equation with delay, Comput. Methods Differ. Equ., 1-14, DOI: 10.22034/cmde.2020.35546.1608.
  15. [15]  Song, H.T. and Xue, D.S. (2014), Blow up in a Nonlinear Viscoelastic Wave Equation with Strong Damping, Nonlinear Analysis, 109, 245-251. http://dx.doi.org/10.1016/j.na.2014.06.012.
  16. [16]  Song, H.T. and Zhong, C.K. (2010), Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Analysis: Real World Applications, 11, 3877-3883.\\ http://dx.doi.org/10.1016/j.nonrwa.2010.02.015.
  17. [17] Zennir, K. (2013), Exponential growth of solutions with $L_{p}$-norm of a nonlinear viscoelastic hyperbolic equation, J. Nonlinear Sci. Appl., 6, 252-262.