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Nonexistence of Global Solutions for the Strongly Damped Wave Equation with Variable Coefficients

Year 2022, Volume: 5 Issue: 2, 51 - 56, 30.06.2022
https://doi.org/10.32323/ujma.1062771
https://izlik.org/JA44FL38AH

Abstract

In this work, we deal with the wave equation with variable coefficients. Under proper conditions on variable coefficients, we prove the nonexistence of global solutions.

References

  • [1] V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differ. Equ., 109(2) (1994), 295-308.
  • [2] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = Au+F(u), Trans. Am. Math. Soc., 192 (1974), 1-21.
  • [3] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.
  • [4] S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachrichten, 231 (2001), 105-111.
  • [5] X. Runzhang, S. Jihong, Some generalized results for global well-posedness for wave equations with damping and source terms, Math. Comput. Simul., 80 (2009), 804-807.
  • [6] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.
  • [7] S. Yu, On the strongly damped wave equation with nonlinear damping and source terms, Electron. J. Qual. Theory Differ. Equ., 39 (2009), 1-18.
  • [8] S. Gerbi, B. Said-Houari, Exponential decay for solutions to semilinear damped wave equation, Discrete. Cont. Dyn. - S, 5(3) (2012), 559-566.
  • [9] X. Zheng, Y. Shang, X. Peng, Blow up of solutions for a nonlinear Petrovsky type equation with time-dependent coefficients, Acta Math. Appl. Sin., 36(4) (2020), 836-846.
  • [10] R. A. Adams, J. J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003.
  • [11] E. Pis¸kin, B. Okutmus¸tur, An Introduction to Sobolev Spaces, Bentham Science, 2021.
  • [12] E. Pis¸kin, Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms, Open Math., 13 (2015), 408-420.

Year 2022, Volume: 5 Issue: 2, 51 - 56, 30.06.2022
https://doi.org/10.32323/ujma.1062771
https://izlik.org/JA44FL38AH

Abstract

References

  • [1] V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differ. Equ., 109(2) (1994), 295-308.
  • [2] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = Au+F(u), Trans. Am. Math. Soc., 192 (1974), 1-21.
  • [3] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.
  • [4] S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachrichten, 231 (2001), 105-111.
  • [5] X. Runzhang, S. Jihong, Some generalized results for global well-posedness for wave equations with damping and source terms, Math. Comput. Simul., 80 (2009), 804-807.
  • [6] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.
  • [7] S. Yu, On the strongly damped wave equation with nonlinear damping and source terms, Electron. J. Qual. Theory Differ. Equ., 39 (2009), 1-18.
  • [8] S. Gerbi, B. Said-Houari, Exponential decay for solutions to semilinear damped wave equation, Discrete. Cont. Dyn. - S, 5(3) (2012), 559-566.
  • [9] X. Zheng, Y. Shang, X. Peng, Blow up of solutions for a nonlinear Petrovsky type equation with time-dependent coefficients, Acta Math. Appl. Sin., 36(4) (2020), 836-846.
  • [10] R. A. Adams, J. J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003.
  • [11] E. Pis¸kin, B. Okutmus¸tur, An Introduction to Sobolev Spaces, Bentham Science, 2021.
  • [12] E. Pis¸kin, Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms, Open Math., 13 (2015), 408-420.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Erhan Pişkin 0000-0001-6587-4479

Ayşe Fidan

Submission Date January 25, 2022
Acceptance Date May 12, 2022
Publication Date June 30, 2022
DOI https://doi.org/10.32323/ujma.1062771
IZ https://izlik.org/JA44FL38AH
Published in Issue Year 2022 Volume: 5 Issue: 2

Cite

APA Pişkin, E., & Fidan, A. (2022). Nonexistence of Global Solutions for the Strongly Damped Wave Equation with Variable Coefficients. Universal Journal of Mathematics and Applications, 5(2), 51-56. https://doi.org/10.32323/ujma.1062771
AMA 1.Pişkin E, Fidan A. Nonexistence of Global Solutions for the Strongly Damped Wave Equation with Variable Coefficients. Univ. J. Math. Appl. 2022;5(2):51-56. doi:10.32323/ujma.1062771
Chicago Pişkin, Erhan, and Ayşe Fidan. 2022. “Nonexistence of Global Solutions for the Strongly Damped Wave Equation With Variable Coefficients”. Universal Journal of Mathematics and Applications 5 (2): 51-56. https://doi.org/10.32323/ujma.1062771.
EndNote Pişkin E, Fidan A (June 1, 2022) Nonexistence of Global Solutions for the Strongly Damped Wave Equation with Variable Coefficients. Universal Journal of Mathematics and Applications 5 2 51–56.
IEEE [1]E. Pişkin and A. Fidan, “Nonexistence of Global Solutions for the Strongly Damped Wave Equation with Variable Coefficients”, Univ. J. Math. Appl., vol. 5, no. 2, pp. 51–56, June 2022, doi: 10.32323/ujma.1062771.
ISNAD Pişkin, Erhan - Fidan, Ayşe. “Nonexistence of Global Solutions for the Strongly Damped Wave Equation With Variable Coefficients”. Universal Journal of Mathematics and Applications 5/2 (June 1, 2022): 51-56. https://doi.org/10.32323/ujma.1062771.
JAMA 1.Pişkin E, Fidan A. Nonexistence of Global Solutions for the Strongly Damped Wave Equation with Variable Coefficients. Univ. J. Math. Appl. 2022;5:51–56.
MLA Pişkin, Erhan, and Ayşe Fidan. “Nonexistence of Global Solutions for the Strongly Damped Wave Equation With Variable Coefficients”. Universal Journal of Mathematics and Applications, vol. 5, no. 2, June 2022, pp. 51-56, doi:10.32323/ujma.1062771.
Vancouver 1.Pişkin E, Fidan A. Nonexistence of Global Solutions for the Strongly Damped Wave Equation with Variable Coefficients. Univ. J. Math. Appl. [Internet]. 2022 June 1;5(2):51-6. Available from: https://izlik.org/JA44FL38AH

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