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Year 2020, Volume: 3 Issue: 1, 97 - 101, 15.12.2020

Erhan Pişkin Hazal Yüksekkaya

Abstract

References

  • 1 S. Antontsev, Wave equation with $p(x; t)$-Laplacian and damping term: blow-up of solutions, C. R. Mecanique, 339 (2011), 751-755.
  • 2 S. Antontsev, Wave equation with $p(x; t)$-Laplacian and damping term: existence and blow-up, Differential Equations Appl., 3 (2011), 503-525.
  • 3 Y. Chen, S. Levine, M. Rao, Variable Exponent, Linear Growth Functionals in Image Restoration, SIAM Journal on Applied Mathematics, 66 (2006), 1383-1406.
  • 4 L. Diening, P. Hasto, P. Harjulehto, M.M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, 2011.
  • 5 X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces $Wk;p(x) ()$, J. Math. Anal. Appl., 263 (2001), 749-760.
  • 6 M. Kafini and S. A. Messaoudi, A blow-up result in a nonlinear wave equation with delay, Mediterr. J. Math., 13 (2016), 237-247.
  • 7 O. Kovacik , J. Rakosnik, On spaces $Lp(x) ()$ and $Wk;p(x) ()$, Czech. Math. J., 41(116) (1991), 592-618.
  • 8 S. A. Messaoudi and M. Kafini, On the decay and global nonexistence of solutions to a damped wave equation with variable-exponent nonlinearity and delay, Ann. Pol. Math., 122.1 (2019), doi: 10.4064/ap180524-31-10.
  • 9 S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
  • 10 E. Pişkin, Sobolev Spaces, Seçkin Publishing, 2017 (in Turkish).
  • 11 S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Mag. J. Sci., 41(6) (1921), 744-746.

Nonexistence of Solutions of a Delayed Wave Equation with Variable-Exponents

Year 2020, Volume: 3 Issue: 1, 97 - 101, 15.12.2020

Erhan Pişkin Hazal Yüksekkaya

Abstract

In this paper, we deal with a nonlinear Timoshenko equation with delay term and variable exponents. Under suitable conditions, we prove the blow-up of solutions in a finite time. Our results are more general than the earlier results. Time delays arise in many applications, for instance, it appears in physical, chemical, biological, thermal and economic phenomena. Also, delay is source of instability, a small delay can destabilize a system which is uniformly asymptotically stable. Several physical phenomena such as flows of electro-rheological fluids or fluids with temperature-dependent viscosity, nonlinear viscoelasticity, filtration processes through a porous media and image processing are modelled by equations with variable exponents.

Keywords

Blow up Timoshenko equation Delay term

References

  • 1 S. Antontsev, Wave equation with $p(x; t)$-Laplacian and damping term: blow-up of solutions, C. R. Mecanique, 339 (2011), 751-755.
  • 2 S. Antontsev, Wave equation with $p(x; t)$-Laplacian and damping term: existence and blow-up, Differential Equations Appl., 3 (2011), 503-525.
  • 3 Y. Chen, S. Levine, M. Rao, Variable Exponent, Linear Growth Functionals in Image Restoration, SIAM Journal on Applied Mathematics, 66 (2006), 1383-1406.
  • 4 L. Diening, P. Hasto, P. Harjulehto, M.M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, 2011.
  • 5 X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces $Wk;p(x) ()$, J. Math. Anal. Appl., 263 (2001), 749-760.
  • 6 M. Kafini and S. A. Messaoudi, A blow-up result in a nonlinear wave equation with delay, Mediterr. J. Math., 13 (2016), 237-247.
  • 7 O. Kovacik , J. Rakosnik, On spaces $Lp(x) ()$ and $Wk;p(x) ()$, Czech. Math. J., 41(116) (1991), 592-618.
  • 8 S. A. Messaoudi and M. Kafini, On the decay and global nonexistence of solutions to a damped wave equation with variable-exponent nonlinearity and delay, Ann. Pol. Math., 122.1 (2019), doi: 10.4064/ap180524-31-10.
  • 9 S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
  • 10 E. Pişkin, Sobolev Spaces, Seçkin Publishing, 2017 (in Turkish).
  • 11 S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Mag. J. Sci., 41(6) (1921), 744-746.
There are 11 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Erhan Pişkin

Hazal Yüksekkaya

Publication Date December 15, 2020
Acceptance Date September 30, 2020
Published in Issue Year 2020 Volume: 3 Issue: 1

Cite

APA Pişkin, E., & Yüksekkaya, H. (2020). Nonexistence of Solutions of a Delayed Wave Equation with Variable-Exponents. Conference Proceedings of Science and Technology, 3(1), 97-101.
AMA Pişkin E, Yüksekkaya H. Nonexistence of Solutions of a Delayed Wave Equation with Variable-Exponents. Conference Proceedings of Science and Technology. December 2020;3(1):97-101.
Chicago Pişkin, Erhan, and Hazal Yüksekkaya. “Nonexistence of Solutions of a Delayed Wave Equation With Variable-Exponents”. Conference Proceedings of Science and Technology 3, no. 1 (December 2020): 97-101.
EndNote Pişkin E, Yüksekkaya H (December 1, 2020) Nonexistence of Solutions of a Delayed Wave Equation with Variable-Exponents. Conference Proceedings of Science and Technology 3 1 97–101.
IEEE E. Pişkin and H. Yüksekkaya, “Nonexistence of Solutions of a Delayed Wave Equation with Variable-Exponents”, Conference Proceedings of Science and Technology, vol. 3, no. 1, pp. 97–101, 2020.
ISNAD Pişkin, Erhan - Yüksekkaya, Hazal. “Nonexistence of Solutions of a Delayed Wave Equation With Variable-Exponents”. Conference Proceedings of Science and Technology 3/1 (December 2020), 97-101.
JAMA Pişkin E, Yüksekkaya H. Nonexistence of Solutions of a Delayed Wave Equation with Variable-Exponents. Conference Proceedings of Science and Technology. 2020;3:97–101.
MLA Pişkin, Erhan and Hazal Yüksekkaya. “Nonexistence of Solutions of a Delayed Wave Equation With Variable-Exponents”. Conference Proceedings of Science and Technology, vol. 3, no. 1, 2020, pp. 97-101.
Vancouver Pişkin E, Yüksekkaya H. Nonexistence of Solutions of a Delayed Wave Equation with Variable-Exponents. Conference Proceedings of Science and Technology. 2020;3(1):97-101.
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