Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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GLOBAL NONEXISTENCE FOR THE WAVE EQUATION WITH BOUNDARY VARIABLE EXPONENT NONLINEARITIES

  • Ha, Tae Gab (Department of Mathematics and Institute of Pure and Applied Mathematics Jeonbuk National University) ;
  • Park, Sun-Hye (Office for Education Accreditation Pusan National University)
  • Received : 2021.05.30
  • Accepted : 2021.09.27
  • Published : 2022.01.01
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Abstract

This paper deals with a nonlinear wave equation with boundary damping and source terms of variable exponent nonlinearities. This work is devoted to prove a global nonexistence of solutions for a nonlinear wave equation with nonnegative initial energy as well as negative initial energy.

Keywords

Acknowledgement

T. G. Ha was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1I1A3A01051714). S.-H. Park was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2020R1I1A3066250).

References

  1. R. Aboulaich, D. Meskine, and A. Souissi, New diffusion models in image processing, Comput. Math. Appl. 56 (2008), no. 4, 874-882. https://doi.org/10.1016/j.camwa.2008.01.017
  2. S. Antontsev, Wave equation with p(x, t)-Laplacian and damping term: existence and blow-up, Differ. Equ. Appl. 3 (2011), no. 4, 503-525. https://doi.org/10.7153/dea03-32
  3. S. Antontsev and S. Shmarev, Evolution PDEs with nonstandard growth conditions, Atlantis Studies in Differential Equations, 4, Atlantis Press, Paris, 2015. https://doi.org/10.2991/978-94-6239-112-3
  4. L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8
  5. X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl. 339 (2008), no. 2, 1395-1412. https://doi.org/10.1016/j.jmaa.2007.08.003
  6. V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations 109 (1994), no. 2, 295-308. https://doi.org/10.1006/jdeq.1994.1051
  7. S. Ghegal, I. Hamchi, and S. A. Messaoudi, Global existence and stability of a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal. 99 (2020), no. 8, 1333-1343. https://doi.org/10.1080/00036811.2018.1530760
  8. T. G. Ha, Blow-up for semilinear wave equation with boundary damping and source terms, J. Math. Anal. Appl. 390 (2012), no. 1, 328-334. https://doi.org/10.1016/j.jmaa.2012.01.037
  9. T. G. Ha, Blow-up for wave equation with weak boundary damping and source terms, Appl. Math. Lett. 49 (2015), 166-172. https://doi.org/10.1016/j.aml.2015.05.003
  10. H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = -Au + &3x1D4D5;(u), Trans. Amer. Math. Soc. 192 (1974), 1-21. https://doi.org/10.2307/1996814
  11. H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974), 138-146. https://doi.org/10.1137/0505015
  12. H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal. 137 (1997), no. 4, 341-361. https://doi.org/10.1007/s002050050032
  13. H. A. Levine and G. Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. Soc. 129 (2001), no. 3, 793-805. https://doi.org/10.1090/S0002-9939-00-05743-9
  14. S. A. Messaoudi, J. H. Al-Smail, and A. A. Talahmeh, Decay for solutions of a nonlinear damped wave equation with variable-exponent nonlinearities, Comput. Math. Appl. 76 (2018), no. 8, 1863-1875. https://doi.org/10.1016/j.camwa.2018.07.035
  15. S. A. Messaoudi and A. A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Methods Appl. Sci. 40 (2017), no. 18, 6976-6986. https://doi.org/10.1002/mma.4505
  16. S. A. Messaoudi, A. A. Talahmeh, and J. H. Al-Smail, Nonlinear damped wave equation: existence and blow-up, Comput. Math. Appl. 74 (2017), no. 12, 3024-3041. https://doi.org/10.1016/j.camwa.2017.07.048
  17. L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3-4, 273-303. https://doi.org/10.1007/BF02761595
  18. P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations 150 (1998), no. 1, 203-214. https://doi.org/10.1006/jdeq.1998.3477
  19. A. Rahmoune, Existence and asymptotic stability for the semilinear wave equation with variable-exponent nonlinearities, J. Math. Phys. 60 (2019), no. 12, 122701, 23 pp. https://doi.org/10.1063/1.5089879
  20. M. Ruzicka, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/BFb0104029
  21. G. Todorova and E. Vitillaro, Blow-up for nonlinear dissipative wave equations in ℝn, J. Math. Anal. Appl. 303 (2005), no. 1, 242-257. https://doi.org/10.1016/j.jmaa.2004.08.039
  22. E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J. 44 (2002), no. 3, 375-395. https://doi.org/10.1017/S0017089502030045
  23. E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149 (1999), no. 2, 155-182. https://doi.org/10.1007/s002050050171