Skip to main content
SpringerLink
Log in

Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this work, we determine the critical exponent for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower-order terms, when these terms make both equations in some sense “parabolic-like.” For the blow-up result, the test functions method is applied, while for the global existence (in time) results, we use \(L^2\)–\(L^2\) estimates with additional \(L^1\) regularity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agemi, R., Kurokawa, Y., Takamura, H.: Critical curve for \(p\)–\(q\) systems of nonlinear wave equations in three space dimensions. J. Differ. Equ. 167(1), 87–133 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, W., Reissig, M.: Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D. Math. Methods Appl. Sci. 42(2), 667–709 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. D’Abbicco M.: A note on a weakly coupled system of structurally damped waves. Discrete and Continuous Dynamical systems, differential equations and applications. In: 10th AIMS Conference. Supplement, pp. 320–329 (2015). https://doi.org/10.3934/proc.2015.0320

  4. D’Abbicco, M.: The threshold of effective damping for semilinear wave equations. Math. Methods Appl. Sci. 38(6), 1032–1045 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. D’Abbicco M., Lucente S.: NLWE with a special scale invariant damping in odd space dimension. Discrete and Continuous Dynamical systems, differential equations and applications. 10th AIMS Conference, Supplement, 312–319 (2015). https://doi.org/10.3934/proc.2015.0312

  6. D’Abbicco, M., Lucente, S., Reissig, M.: A shift in the Strauss exponent for semilinear wave equations with a not effective damping. J. Differ. Equ. 259(10), 5040–5073 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. D’Abbicco M., Palmieri A.: \(L^p\)-\(L^q\) estimates on the conjugate line for semilinear critical dissipative Klein–Gordon equations (under review)

  8. Del Santo, D.: Global existence and blow-up for a hyperbolic system in three space dimensions. Rend. Istit. Mat. Univ. Trieste 29(1/2), 115–140 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Del Santo, D., Mitidieri, E.: Blow-up of solutions of a hyperbolic system: the critical case. Differ. Equ. 34(9), 1157–1163 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Del Santo, D., Georgiev, V., Mitidieri, E.: Global existence of the solutions and formation of singularities for a class of hyperbolic systems. In: Colombini, F., Lerner, N. (eds.) Geometrical Optics and Related Topics Progress in Nonlinear Differential Equations and Their Applications, vol. 32. Birkhäuser, Boston, MA (1997). https://doi.org/10.1007/978-1-4612-2014-5_7

    Chapter  Google Scholar 

  11. Egorov, Y.V., Galaktionov, V.A., Kondratiev, V.A., Pohozaev, S.I.: On the necessary conditions of global existence to a quasilinear inequality in the half-space. C. R. Acad. Sci. Paris Sér. I Math. 330(2), 93–98 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Galaktionov, V.A., Mitidieri, E., Pohozaev, S.I.: Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL (2015)

    MATH  Google Scholar 

  13. Georgiev, V., Lindblad, H., Sogge, C.D.: Weighted Strichartz estimates and global existence for semilinear wave equations. Am. J. Math. 119(6), 1291–1319 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Georgiev, V., Takamura, H., Zhou, Y.: The lifespan of solutions to nonlinear systems of a high-dimensional wave equation. Nonlinear Anal. 64(10), 2215–2250 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Glassey, R.T.: Existence in the large for \(\square u = F(u)\) in two space dimensions. Math. Z. 178(2), 233–261 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Glassey, R.T.: Finite-time blow-up for solutions of nonlinear wave equations. Math. Z. 177(3), 323–340 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ikeda, M., Sobajima, M.: Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data. Math. Ann. (2018). https://doi.org/10.1007/s00208-018-1664-1

    Article  MathSciNet  MATH  Google Scholar 

  18. Ikehata, R., Tanizawa, K.: Global existence of solutions for semilinear damped wave equations in \(\mathbf{R}^N\) with noncompactly supported initial data. Nonlinear Anal. 61(7), 1189–1208 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jiao, H., Zhou, Z.: An elementary proof of the blow-up for semilinear wave equation in high space dimensions. J. Differ. Equ. 189(2), 355–365 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. John, F.: Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscr. Math. 28, 235 (1979). https://doi.org/10.1007/BF01647974

    Article  MathSciNet  MATH  Google Scholar 

  21. Kato, M., Sakuraba, M.: Global existence and blow-up for semilinear damped wave equations in three space dimensions (preprint). arXiv:1807.04327v1

  22. Kato, T.: Blow-up of solutions of some nonlinear hyperbolic equations. Commun. Pure Appl. Math. 33(4), 501–505 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kurokawa, Y.: The lifespan of radially symmetric solutions to nonlinear systems of odd dimensional wave equations. Tsukuba J. Math. 60(7), 1239–1275 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Kurokawa, Y., Takamura, H.: A weighted pointwise estimate for two dimensional wave equations and its applications to nonlinear systems. Tsukuba J. Math. 27(2), 417–448 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kurokawa, Y., Takamura, H., Wakasa, K.: The blow-up and lifespan of solutions to systems of semilinear wave equation with critical exponents in high dimensions. Differ. Integral Equ. 25(3/4), 363–382 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Lai, N.A.: Weighted \(L^2-L^2\) estimates for wave equation and its applications (preprint). arXiv:1807.05109v1

  27. Lai, N.A., Takamura, H., Wakasa, K.: Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent. J. Differ. Equ. 263(9), 5377–5394 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lai, N.A., Zhou, Y.: An elementary proof of Strauss conjecture. J. Funct. Anal. 267(5), 1364–1381 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lindblad, H., Sogge, C.D.: Long-time existence for small amplitude semilinear wave equations. Am. J. Math. 118(5), 1047–1135 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mitidieri, E., Pohozaev, S.I.: Absence of global positive solutions of quasilinear elliptic inequalities. Dokl. Akad. Nauk 359(4), 456–460 (1998)

    MathSciNet  MATH  Google Scholar 

  31. Mitidieri, E., Pohozaev, S.I.: Absence of positive solutions for systems of quasilinear elliptic equations and inequalities in \(\mathbb{R}^N\). Dokl. Akad. Nauk 366(1), 13–17 (1999)

    MathSciNet  Google Scholar 

  32. Mitidieri, E., Pohozaev, S.I.: A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. Proc. Steklov Inst. Math. 234, 1–375 (2001)

    MATH  Google Scholar 

  33. Mitidieri, E., Pohozaev, S.I.: Nonexistence of weak solutions for some degenerate and singular hyperbolic problems on \(\mathbb{R}^{n+1}_+\). Proc. Steklov Inst. Math. 232, 240–259 (2001)

    MATH  Google Scholar 

  34. Mitidieri, E., Pohozaev, S.I.: Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on \(\mathbb{R}^n\). J. Evol. Equ. 1(2), 189–220 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mohammed, Djaouti A., Reissig, M.: Weakly coupled systems of semilinear effectively damped waves with time-dependent coefficient, different power nonlinearities. Nonlinear Anal. 175, 28–55 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mohammed, Djaouti A.: On the benefit of different additional regularity for the weakly coupled systems of semilinear effectively damped waves. Mediterr. J. Math. 15, 115 (2018). https://doi.org/10.1007/s00009-018-1173-1

    Article  MathSciNet  MATH  Google Scholar 

  37. Nunes do Nascimento, W., Palmieri, A., Reissig, M.: Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation. Math. Nachr. 290(11/12), 1779–1805 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Narazaki T.: Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Discrete and Continuous Dynamical systems, differential equations and applications. In: 7th AIMS Conference. Supplement, pp. 592–601 (2009). https://doi.org/10.3934/proc.2009.2009.592

  39. Narazaki, T.: Global solutions to the Cauchy problem for a system of damped wave equations. Differ. Integral Equ. 24(5/6), 569–600 (2011)

    MathSciNet  MATH  Google Scholar 

  40. Nishihara, K.: Asymptotic behavior of solutions for a system of semilinear heat equations and the corresponding damped wave system. Osaka J. Math. 49(2), 331–348 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Nishihara, K., Wakasugi, Y.: Critical exponent for the Cauchy problem to the weakly coupled damped wave system. Nonlinear Anal. 108, 249–259 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Nishihara, K., Wakasugi, Y.: Global existence of solutions for a weakly coupled system of semilinear damped wave equations. J. Differ. Equ. 259(8), 4172–4201 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Nishihara, K., Wakasugi, Y.: Critical exponents for the Cauchy problem to the system of wave equations with time or space dependent damping. Bull. Inst. Math. Acad. Sin. (N. S.) 10(3), 283–309 (2015)

    MathSciNet  MATH  Google Scholar 

  44. Ogawa, T., Takeda, H.: Global existence of solutions for a system of nonlinear damped wave equations. Differ. Integral Equ. 23(7/8), 635–657 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Ogawa, T., Takeda, H.: Large time behavior of solutions for a system of nonlinear damped wave equations. J. Differ. Equ. 251(11), 3090–3113 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Palmieri, A.: Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces. J. Math. Anal. Appl. 461(2), 1215–1240 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  47. Palmieri A.: Global existence results for a semilinear wave equation with scale-invariant damping and mass in odd space dimension. In: D’Abbicco, M., et al. (eds.), New Tools for Nonlinear PDEs and Application, Trends in Mathematics, Springer, Switzerland AG (2019). (to appear)

  48. Palmieri, A.: A global existence result for a semilinear wave equation with scale-invariant damping and mass in even space dimension. Math. Methods Appl. Sci. 1–27 (2019). https://doi.org/10.1002/mma.5542

    Article  MathSciNet  MATH  Google Scholar 

  49. Palmieri, A.: A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms (preprint). arXiv:1812.06588

  50. Palmieri, A., Reissig, M.: Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation. II. Math. Nachr. 291(11/12), 1859–1892 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  51. Palmieri, A., Reissig, M.: A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass. J. Differ. Equ. (2018). https://doi.org/10.1016/j.jde.2018.07.061

    Article  MATH  Google Scholar 

  52. Palmieri, A., Tu, Z.: Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity. J. Math. Anal. Appl. 470, 447–469 (2019). https://doi.org/10.1016/j.jmaa.2018.10.015

    Article  MathSciNet  MATH  Google Scholar 

  53. Schaeffer, J.: The equation \(u_{tt}-\Delta u = |u|^p\) for the critical value of \(p\). Proc. R. Soc. Edinb. Sect. A. 101(1/2), 31–44 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  54. Sideris, T.C.: Nonexistence of global solutions to semilinear wave equations in high dimensions. J. Differ. Equ. 52(3), 378–406 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  55. Strauss, W.A.: Nonlinear scattering theory at low energy. J. Funct. Anal. 41(1), 110–133 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  56. Sun, F., Wang, M.: Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping. Nonlinear Anal. 66(12), 2889–2910 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  57. Todorova, G., Yordanov, B.: Critical exponent for a nonlinear wave equation with damping. J. Differ. Equ. 174(2), 464–489 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  58. Tu, Z., Lin, J.: A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent (preprint). arXiv:1709.00866v2

  59. Tu, Z., Lin, J.: Life-span of semilinear wave equations with scale-invariant damping: Critical Strauss exponent case (preprint). arXiv:1711.00223

  60. Wakasa, K.: The lifespan of solutions to semilinear damped wave equations in one space dimension. Commun. Pure Appl. Anal. 15(4), 1265–1283 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  61. Wakasugi, Y.: Critical exponent for the semilinear wave equation with scale invariant damping. In: Ruzhansky, M., Turunen, V. (eds.) Fourier Analysis Trends in Mathematics. Birkhäuser, Cham (2014). https://doi.org/10.1007/978-3-319-02550-6_19

    Chapter  Google Scholar 

  62. Yordanov, B.T., Zhang, Q.S.: Finite time blow up for critical wave equations in high dimensions. J. Funct. Anal. 231(2), 361–374 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  63. Zhang, Q.S.: A blow-up result for a nonlinear wave equation with damping: the critical case. C. R. Acad. Sci. Paris Sér. I Math. 333(2), 109–114 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  64. Zhou, Y.: Cauchy problem for semilinear wave equations in four space dimensions with small initial data. J. Partial Differ. Equ. 8(2), 135–144 (1995)

    MathSciNet  MATH  Google Scholar 

  65. Zhou, Y.: Blow up of solutions to semilinear wave equations with critical exponent in high dimensions. Chin. Ann. Math. Ser. B 28(2), 205–212 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The Ph.D. study of the first author is supported by Sächsiches Landesgraduiertenstipendium. The second author is member of the Gruppo Nazionale per L’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Instituto Nazionale di Alta Matematica (INdAM). The second author is supported by the University of Pisa, Project PRA 2018 49. This work was partially written while the second author was a Ph.D. student at TU Freiberg. Finally, the authors thank their supervisor Michael Reissig for the suggestions in the preparation of the final version and the anonymous referees for carefully reading the paper.

Author information

Authors and Affiliations

  1. Institute of Applied Analysis, Faculty for Mathematics and Computer Science, Technical University Bergakademie Freiberg, Prüferstraße 9, 09596, Freiberg, Germany

    Wenhui Chen

  2. Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Alessandro Palmieri

Authors
  1. Wenhui Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessandro Palmieri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandro Palmieri.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, W., Palmieri, A. Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part. Z. Angew. Math. Phys. 70, 67 (2019). https://doi.org/10.1007/s00033-019-1112-4

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-019-1112-4

Mathematics Subject Classification

Keywords

Search

Navigation