Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

BLOW-UP OF SOLUTIONS FOR WAVE EQUATIONS WITH STRONG DAMPING AND VARIABLE-EXPONENT NONLINEARITY

  • Park, Sun-Hye (Office for Education Accreditation Pusan National University)
  • Received : 2020.04.18
  • Accepted : 2020.07.21
  • Published : 2021.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we consider the following strongly damped wave equation with variable-exponent nonlinearity utt(x, t) - ∆u(x, t) - ∆ut(x, t) = |u(x, t)|p(x)-2u(x, t), where the exponent p(·) of nonlinearity is a given measurable function. We establish finite time blow-up results for the solutions with non-positive initial energy and for certain solutions with positive initial energy. We extend the previous results for strongly damped wave equations with constant exponent nonlinearity to the equations with variable-exponent nonlinearity.

Keywords

Acknowledgement

This research was supported by 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/dea-03-32
  3. S. Antontsev and S. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math. 234 (2010), no. 9, 2633-2645. https://doi.org/10.1016/j.cam.2010.01.026
  4. 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
  5. J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 112, 473-486. https://doi.org/10.1093/qmath/28.4.473
  6. M. M. Cavalcanti, V. N. Domingos Cavalcanti, and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations 203 (2004), no. 1, 119-158. https://doi.org/10.1016/j.jde.2004.04.011
  7. 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
  8. D. E. Edmunds and J. Rakosnik, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267-293. https://doi.org/10.4064/sm-143-3-267-293
  9. D. E. Edmunds and J. Rakosnik, Sobolev embeddings with variable exponent. II, Math. Nachr. 246/247 (2002), 53-67. https://doi.org/10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T
  10. J. A. Esquivel-Avila, The dynamics of a nonlinear wave equation, J. Math. Anal. Appl. 279 (2003), no. 1, 135-150. https://doi.org/10.1016/S0022-247X(02)00701-1
  11. X. Fan and D. Zhao, On the spaces Lp(x) (Ω) and Wm,p(x) (Ω), J. Math. Anal. Appl. 263 (2001), no. 2, 424-446. https://doi.org/10.1006/jmaa.2000.7617
  12. F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincare Anal. Non Lineaire 23 (2006), no. 2, 185-207. https://doi.org/10.1016/j.anihpc.2005.02.007
  13. 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
  14. 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
  15. R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal. 27 (1996), no. 10, 1165-1175. https://doi.org/10.1016/0362-546X(95)00119-G
  16. O. Kovacik and J. Rakosnik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41(116) (1991), no. 4, 592-618. https://doi.org/10.21136/CMJ.1991.102493
  17. H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = -Au + F(u), Trans. Amer. Math. Soc. 192 (1974), 1-21. https://doi.org/10.2307/1996814
  18. S. Lian, W. Gao, C. Cao, and H. Yuan, Study of the solutions to a model porous medium equation with variable exponent of nonlinearity, J. Math. Anal. Appl. 342 (2008), no. 1, 27-38. https://doi.org/10.1016/j.jmaa.2007.11.046
  19. J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Second ed. Dunod, Paris, 2002.
  20. S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachr. 231 (2001), 105-111. https://doi.org/10.1002/1522-2616(200111)231:1<105::aid-mana105>3.0.co;2-i
  21. S. A. Messaoudi, Global nonexistence in a nonlinearly damped wave equation, Appl. Anal. 80 (2001), no. 3-4, 269-277. https://doi.org/10.1080/00036810108840993
  22. S. A. Messaoudi, On the decay of solutions of a damped quasilinear wave equation with variable-exponent nonlinearities, Math. Meth. Appl. Sci. 43 (2020), 5114-5126. https://doi.org/10.1002/mma.6254
  23. 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
  24. S. A. Messaoudi and A. A. Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal. 96 (2017), no. 9, 1509-1515. https://doi.org/10.1080/00036811.2016.1276170
  25. 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
  26. 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
  27. D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968), 148-172. https://doi.org/10.1007/BF00250942
  28. G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canadian J. Math. 32 (1980), no. 3, 631-643. https://doi.org/10.4153/CJM-1980-049-5