Mathematische Annalen

, Volume 366, Issue 1–2, pp 667–694 | Cite as

Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations

Article

Abstract

This paper investigates the combined effects of two distinctive power-type nonlinear terms (with parameters \(p,q>1\)) in the lifespan of small solutions to semi-linear wave equations. We determine the full region of (pq) to admit global existence of small solutions, at least for spatial dimensions \(n=2, 3\). Moreover, for many (pq) when there is no global existence, we obtain sharp lower bound of the lifespan, which is of the same order as the upper bound of the lifespan.

Mathematics Subject Classification

35L05 35L15 35L71 

References

  1. 1.
    Agemi, R.: Blow-up of solutions to nonlinear wave equations in two space dimensions. Manuscripta Math. 73, 153–162 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bergh, J., Löfström, J.: Interpolation spaces. An introduction. In: Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)Google Scholar
  3. 3.
    Fang, D., Wang, C.: Weighted Strichartz estimates with angular regularity and their applications. Forum Math. 23, 181–205 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Georgiev, V., Lindblad, H., Sogge, C.D.: Weighted Strichartz estimates and global existence for semilinear wave equations. Am. J. Math. 119, 1291–1319 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Glassey, R.T.: Finite-time blow-up for solutions of nonlinear wave equations. Math. Z. 177, 323–340 (1981)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Glassey, R.T.: Existence in the large for \(\Box u=F(u)\) in two space dimensions. Math. Z. 178, 233–261 (1981)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Han, W., Zhou, Y.: Blow up for some semilinear wave equations in multi-space dimensions. Commun. Partial Differ. Equ. 39, 651–665 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hidano, K.: Small data scattering for wave equations with super critical nonlinearity. In: Proceedings of the 23rd Sapporo Symposium on Partial Differential Equations (Yoshikazu Giga, Edt.), pp. 23–30 (1998). http://eprints3.math.sci.hokudai.ac.jp/1232/1/53
  9. 9.
    Hidano, K., Metcalfe, J., Smith, H.F., Sogge, C.D., Zhou, Y.: On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles. Trans. Am. Math. Soc. 362, 2789–2809 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hidano, K., Tsutaya, K.: Global existence and asymptotic behavior of solutions for nonlinear wave equations. Indiana Univ. Math. J. 44, 1273–1305 (1995)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hidano, K., Wang, C., Yokoyama, K.: The Glassey conjecture with radially symmetric data. J. Math. Pures Appl. (9) 98, 518–541 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hoshiro, T.: On weighted \(L^2\) estimates of solutions to wave equations. J. Anal. Math. 72, 127–140 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    John, F.: Blow-up of solutions of nonlinear wave equations in three space dimensions. Man. Math. 28, 235–268 (1979)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    John, F.: Blow-up for quasilinear wave equations in three space dimensions. Commun. Pure Appl. Math. 34, 29–51 (1981)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Katayama, S.: Lifespan of solutions for two space dimensional wave equations with cubic nonlinearity. Commun. Partial Differ. Equ. 26, 205–232 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kato, T.: Blow-up of solutions of some nonlinear hyperbolic equations. Commun. Pure Appl. Math. 33, 501–505 (1980)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Klainerman, S.: Remarks on the global Sobolev inequalities in the Minkowski space \({\mathbb{R}}^{n+1}\). Commun. Pure Appl. Math. 40, 111–117 (1987)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kubo, H., Kubota, K.: Asymptotic behaviors of radially symmetric solutions of \(\Box u=|u|^p\) for super critical values \(p\) in odd space dimensions. Hokkaido Math. J. 24, 287–336 (1995)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kubo, H., Kubota, K.: Asymptotic behaviors of radially symmetric solutions of \(\Box u=|u|^p\) for super critical values \(p\) in even space dimensions. Japan. J. Math. (N.S.) 24, 191–256 (1998)MathSciNetMATHGoogle Scholar
  20. 20.
    Lai, N.A., Zhou, Y.: An elementary proof of Strauss conjecture. J. Funct. Anal. 267, 1364–1381 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Li, T.T., Zhou, Y.: A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions. Indiana Univ. Math. J. 44, 1207–1248 (1995)MathSciNetMATHGoogle Scholar
  22. 22.
    Lindblad, H., Metcalfe, J., Sogge, C.D., Tohaneanu, M., Wang, C.: The Strauss conjecture on Kerr black hole backgrounds. Math. Ann. 359, 637–661 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lindblad, H., Sogge, C.D.: Long-time existence for small amplitude semilinear wave equations. Am. J. Math. 118, 1047–1135 (1996)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rammaha, M.A.: Finite-time blow-up for nonlinear wave equations in high dimensions. Commun. Partial Differ. Equ. 12, 677–700 (1987)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Schaeffer, J.: The equation \(\Box u=|u|^p\) for the critical value \(p\). Proc. Roy. Soc. Edinburgh Sect. A 101, 31–44 (1985)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Schaeffer, J.: Finite-time blow-up for \(u_{tt}-\Delta u=H(u_r,\, u_t)\). Commun. Partial Differ. Equ. 11, 513–543 (1986)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Sideris, T.C.: Global behavior of solutions to nonlinear wave equations in three dimensions. Commun. Partial Differ. Equ. 8, 1291–1323 (1983)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Sideris, T.C.: Nonexistence of global solutions to semilinear wave equations in high dimensions. J. Differ. Equ. 52, 378–406 (1984)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Smith, H.F., Sogge, C.D., Wang, C.: Strichartz estimates for Dirichlet-wave equations in two dimensions with applications. Trans. Am. Math. Soc. 364, 3329–3347 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Strauss, W.A.: Nonlinear scattering theory at low energy. J. Funct. Anal. 41, 110–133 (1981)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Takamura, H.: Improved Kato’s lemma on ordinary differential inequality and its application to semilinear wave equations. Nonlinear Anal. 125, 227–240 (2015)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Takamura, H., Wakasa, K.: The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions. J. Differ. Equ. 251, 1157–1171 (2011)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tataru, D.: Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation. Trans. Am. Math. Soc. 353, 795–807 (2001)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Tzvetkov, N.: Existence of global solutions to nonlinear massless Dirac system and wave equation with small data. Tsukuba J. Math. 22, 193–211 (1998)MathSciNetMATHGoogle Scholar
  35. 35.
    Wang, C.: The Glassey conjecture on asymptotically flat manifolds. Trans. Am. Math. Soc. 367, 7429–7451 (2015)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Wang, C.: The Glassey conjecture for nontrapping obstacles. J. Differ. Equ. 259, 510–530 (2015)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Wang, C.: Long time existence for semilinear wave equations on asymptotically flat space-times. arXiv:1504.05652
  38. 38.
    Yordanov, B.T., Zhang, Q.S.: Finite time blow up for critical wave equations in high dimensions. J. Funct. Anal. 231, 361–374 (2006)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Zhou, Y.: Cauchy problem for semilinear wave equations in four space dimensions with small initial data. J. Partial Differ. Equ. 8, 135–144 (1995)MathSciNetMATHGoogle Scholar
  40. 40.
    Zhou, Y.: Blow up of solutions to the Cauchy problem for nonlinear wave equations. Chin. Ann. Math. Ser. B 22, 275–280 (2001)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Zhou, Y.: Blow up of solutions to semilinear wave equations with critical exponent in high dimensions. Chin. Ann. Math. Ser. B 28, 205–212 (2007)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Zhou, Y., Han, W.: Blow up of solutions to semilinear wave equations with variable coefficients and boundary. J. Math. Anal. Appl. 374, 585–601 (2011)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Zhou, Y., Han, W.: Life-span of solutions to critical semilinear wave equations. Commun. Partial Differ. Equ. 39, 439–451 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Kunio Hidano
    • 1
  • Chengbo Wang
    • 2
  • Kazuyoshi Yokoyama
    • 3
  1. 1.Department of Mathematics, Faculty of EducationMie UniversityTsuJapan
  2. 2.School of Mathematical SciencesZhejiang UniversityHangzhouChina
  3. 3.Hokkaido University of ScienceSapporoJapan

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