Skip to main content
SpringerLink
Log in

On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations

  • Chapter
New Trends in the Theory of Hyperbolic Equations

Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 159))

Abstract

The aim of this work is twofold. One is to develop an approach for dealing with semilinear wave equations adopted by John [38]. In Section 2, the basis of the argument will be explained in a self-contained way. The other is an application of the approach to systems of wave equations. We shall make use of it to handle the semilinear case in Sections 3,4 and 5, and to consider the quasilinear case in Section 6. In these argument we bring such systems that the single wave components obey different propagation speeds into focus.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Agemi, Blow-up of solutions to nonlinear wave equations in two space dimensions, Manuscripta Math. 73 (1991), 153–162.

    MATH  MathSciNet  Google Scholar 

  2. R. Agemi, Global existence of nonlinear elastic waves, Invent. Math. 142 (2000), 225–250.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Agemi, Y. Kurokawa and H. Takamura, Critical curve for p-q systems of nonlinear wave equations in three space dimensions, J. Differential Equations. 167 (2000), 87–133.

    Article  MathSciNet  Google Scholar 

  4. R. Agemi and H. Takamura, The lifespan of classical solutions to nonlinear wave equations in two space dimensions, Hokkaido Math. J. 21 (1992), 517–542.

    MathSciNet  Google Scholar 

  5. R. Agemi and K. Yokoyama, The null conditions and global existence of solutions to systems of wave equations with different propagation speeds, in “Advances in nonlinear partial differential equations and stochastics” (S. Kawashima and T. Yanagisawa ed.), Series on Adv. in Math. for Appl. Sci., Vol. 48, 43–86, World Scientific, Singapore, 1998.

    Google Scholar 

  6. S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, II, Acta Math. 182 (1999), 1–23.

    MATH  MathSciNet  Google Scholar 

  7. S. Alinhac, The null condition for quasilinear wave equations in two space dimensions, II, Amer. J. Math. 123 (2000), 1071–1101.

    MathSciNet  Google Scholar 

  8. S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), 597–618.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Alinhac, An example of blowup at infinity for a quasilinear wave equation, Astérisque, Autour de l’analyse microlocale 284 (2003), 1–91.

    MATH  Google Scholar 

  10. F. Asakura, Existence of a global solution to a semi-liear wave equation with slowly decreasing initial data in three space dimensions, Comm. Partial Differential Equations 11 (1986), 1459–1487.

    MATH  MathSciNet  Google Scholar 

  11. Y. Choquet-Bruhat, Global existence for solutions of □u = A|u|p, J. Differential Equations 82 (1989) 98–108.

    MATH  MathSciNet  Google Scholar 

  12. D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), 267–282.

    MATH  MathSciNet  Google Scholar 

  13. R. Courant and D. Hilbert, “Methods of Mathematical physics”, Vol.II, Interscience Publ.,1962.

    Google Scholar 

  14. P. D’Ancona, V. Georgiev and H. Kubo, Weighted decay estimates for the wave equation, J. Differential Equations 177 (2001), 146–208.

    MathSciNet  Google Scholar 

  15. J.-M. Delort, D. Fang, R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal. 211 (2004), 288–323.

    Article  MathSciNet  Google Scholar 

  16. M. Di Flaviano, Lower bounds of the life span of classical solutions to a system of semilinear wave equations in two space dimensions, J. Math. Anal. Appl. 281 (2003), 22–45.

    MATH  MathSciNet  Google Scholar 

  17. D. Del Santo, Global existence and blow-up for a hyperbolic system in three space dimensions, Rend. Istit. Math. Univ. Trieste 26 (1997), 115–140.

    Google Scholar 

  18. D. Del Santo, V. Georgiev and E. Mitidieri, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, in: “Geometric optics and related topics” (F. Colombini and N. Lerner ed.), Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, 117–140, Birkhäuser, Boston, 1997.

    Google Scholar 

  19. D. Del Santo and E. Mitidieri, Blow-up of solutions of a hyperbolic system: the critical case, Differential Equations 34 (1998), 1157–1163.

    MathSciNet  Google Scholar 

  20. K. Deng, Nonexistence of global solutions of a nonlinear hyperbolic system, Trans. Amer. Math. Soc. 349 (1997), 1685–1696.

    Article  MATH  MathSciNet  Google Scholar 

  21. V. Georgiev, Existence of global solutions to supercritical semilinear wave equations, Serdica Math. J. 22 (1996), 125–164.

    MATH  MathSciNet  Google Scholar 

  22. V. Georgiev, “Semilinear Hyperbolic equations”, MSJ Memoirs, Vol. 7, Math. Soc. of Japan, Tokyo, 2000.

    Google Scholar 

  23. V. Georgiev, H. Lindblad and C. Sogge, Weighted Strichartz estimate and global existence for semilinear wave equation, Amer. J. Math. 119 (1997), 1291–1319.

    MathSciNet  Google Scholar 

  24. J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation, Math. Z 189 (1985) 487–505.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. R. T. Glassey, Existence in the large for □u = F(u) in two space dimensions, Math. Z. 178 (1981), 233–261.

    Article  MATH  MathSciNet  Google Scholar 

  27. P. Godin, Lifespan of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations 18 (1993), 895–916.

    MATH  MathSciNet  Google Scholar 

  28. K. Hidano, Nonlinear small data scattering for the wave equation inℝ4+1, J. Math. Soc. Japan 50 (1998), 253–292.

    MATH  MathSciNet  Google Scholar 

  29. K. Hidano, Scattering and self-similar solutions for the nonlinear wave equation, Differential Integral Equations 15 (2002), 405–462.

    MATH  MathSciNet  Google Scholar 

  30. K. Hidano, The global existence theorem for quasi-linear wave equations with multiple speeds, Hokkaido Math. J. 33 (2004), 607–636.

    MATH  MathSciNet  Google Scholar 

  31. K. Hidano and K. Tsutaya, Global existence and asymptotic behavior of solutions for nonlinear wave equations, Indiana Univ. Math. J. 44 (1995), 1273–1305.

    Article  MathSciNet  Google Scholar 

  32. L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Lecture Note in Math. 1256 (1987), 214–280.

    MATH  Google Scholar 

  33. A. Hoshiga, The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math. J. 24 (1995), 575–615.

    MATH  MathSciNet  Google Scholar 

  34. A. Hoshiga, The lifespan of solutions to quasilinear hyperbolic systems in two space dimensions, Nonlinear Analysis 42 (2000), 543–560.

    Article  MATH  MathSciNet  Google Scholar 

  35. A. Hoshiga, Existence and blowing up of solutions to systems of quasilinear wave equations in two space dimensions, Preprint.

    Google Scholar 

  36. A. Hoshiga and H. Kubo, Global small amplitude solutions of nonlinear hyperbolic systems with a critical exponent under the null condition, SIAM J. Math. Anal. 31 (2000), 486–513.

    Article  MathSciNet  Google Scholar 

  37. A. Hoshiga and H. Kubo, Global solvability for systems of nonlinear wave equations with multiple speeds in two space dimensions, Differential Integral Equations 17 (2004), 593–622.

    MathSciNet  Google Scholar 

  38. F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235–268.

    Article  MATH  MathSciNet  Google Scholar 

  39. F. John, Blow-up of solutions for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math. 34 (1981), 29–51.

    MATH  MathSciNet  Google Scholar 

  40. F. John, Blow-up of radial soluions of utt = c2(ut) Δu in three space dimensions, Mat. Apl. Comput., V (1985), 3–18.

    Google Scholar 

  41. F. John, Nonlinear wave equations, Formation of singularities, Pitcher lectures in mathematical sciences, Lehigh Univ., American Math. Soc., Providence, RI, 1990.

    Google Scholar 

  42. S. Katayama, Global existence for systems of nonlinear wave equations in two space dimensions, II, Publ. RIMS, Kyoto Univ. 31 (1995), 645–665.

    MATH  MathSciNet  Google Scholar 

  43. S. Katayama, Global and almost global existence for systems of nonlinear wave equations with different propagation speeds, Differential Integral Equations 17 (2004), 1043–1078.

    MathSciNet  Google Scholar 

  44. S. Katayama, Global existence for a class of systems of nonlinear wave equations in three space dimensions, Chinese Ann. Math. Ser. B 25 (2004), 463–482.

    MATH  MathSciNet  Google Scholar 

  45. S. Katayama and A. Matsumura, Sharp lower bound for the lifespan of systems of semilinear wave equations with multiple speeds Preprint.

    Google Scholar 

  46. S. Katayama and K. Yokoyama, Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds, Preprint.

    Google Scholar 

  47. J. Kato and T. Ozawa, Weighted Strichartz estimates and existence of self-similar solutions for semilinear wave equations, Indiana Univ. Math. J. 52 (2003), 1615–1630.

    Article  MathSciNet  Google Scholar 

  48. J. Kato and T. Ozawa, On solutions of the wave equation with homogeneous Cauchy data, Asymptotic Anal. 37 (2004), 93–107.

    MathSciNet  Google Scholar 

  49. J. Kato and T. Ozawa, Weighted Strichartz estimates for the wave equation in even space dimensions, Math. Z. 247 (2004), 747–764.

    Article  MathSciNet  Google Scholar 

  50. J. Kato, M. Nakamura and T. Ozawa, On some generalization of the weighted Strichartz estimates for the wave equation and self-similar solutions to nonlinear wave equations, Preprint.

    Google Scholar 

  51. T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), 191–205.

    Article  Google Scholar 

  52. M. Keel and T. Tao, Endpoints Strichartz estimates, Amer. J. Math. 120 (1998), 955–980.

    MathSciNet  Google Scholar 

  53. S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), 321–332.

    MATH  MathSciNet  Google Scholar 

  54. S. Klainerman, The null condition and global existence to nonlinear wave equations, Lectures in Appl. Math. 23 (1986), 293–326.

    MATH  MathSciNet  Google Scholar 

  55. S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski spaceℝn+1, Comm. Pure Appl. Math. 40 (1987), 111–117.

    MATH  MathSciNet  Google Scholar 

  56. S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 (1996), 307–321.

    Article  MathSciNet  Google Scholar 

  57. M. Kovalyov, Resonance-type behaviour in a system of nonlinear wave equations, J. Differential Equations 77 (1989), 73–83.

    Article  MATH  MathSciNet  Google Scholar 

  58. H. Kubo, Blow-up of solutions to semilinear wave equations with initial data of slow decay in low space dimenions, Differential Integral Equations 7 (1994), 315–321.

    MATH  MathSciNet  Google Scholar 

  59. H. Kubo, On the critical decay and power for semilinear wave equations in odd space dimenions, Discrete Contin. Dynam. Systems 2 (1996), 173–190.

    MATH  MathSciNet  Google Scholar 

  60. H. Kubo and K. Kubota, Asymptotic behaviors of radially symmetric solutions of □u = |u|pfor super critical values p in odd space dimensions, Hokkaido Math. J. 24 (1995), 287–336.

    MathSciNet  Google Scholar 

  61. H. Kubo and K. Kubota, Asymptotic behaviors of radially symmetric solutions of □u = |u|pfor super critical values p in even space dimensions, Japanese J. Math. 24 (1998), 191–256.

    MathSciNet  Google Scholar 

  62. H. Kubo and K. Kubota, Asymptotic behavior of classical solutions to a system of semilinier wave equations in low space dimensions, J. Math. Soc. Japan. 53 (2001), 875–912.

    MathSciNet  Google Scholar 

  63. H. Kubo and K. Kubota, Scattering for systems of semilinier wave equations with different speeds of propagation, Advances in Diff. Eq. 7 (2002), 441–468.

    MathSciNet  Google Scholar 

  64. H. Kubo and K. Kubota, Existence and asymptotic behavior of radially symmetric solutions to a semilinear hyperbolic system in odd space dimensions, Preprint.

    Google Scholar 

  65. H. Kubo and M. Ohta, Critical blowup for systems of semilinear wave equations in low space dimensions, J. Math. Anal. Appl. 240 (1999), 340–360.

    Article  MathSciNet  Google Scholar 

  66. H. Kubo and M. Ohta, Small data blowup for systems of semilinear wave equations with different propagation speeds in three space dimensions, J. Differential Equations 163 (2000), 475–492.

    Article  MathSciNet  Google Scholar 

  67. H. Kubo and M. Ohta, Global existence and blow-up of the classical solutions to systems of semilinear wave equations in three space dimensions, Rend. Istit. Math. Univ. Trieste 31 (2000), 145–168.

    MathSciNet  Google Scholar 

  68. H. Kubo and M. Ohta, On systems of semilinear wave equations with unequal propagation speeds in three space dimensions, to appear in: Funkcialaj Ekvacioj.

    Google Scholar 

  69. H. Kubo and M. Ohta, Blowup for systems of semilinear wave equations in two space dimensions, Preprint.

    Google Scholar 

  70. H. Kubo and K. Tsugawa, Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions, Discrete Contin. Dynam. Systems 9 (2003), 471–482.

    MathSciNet  Google Scholar 

  71. K. Kubota, Existence of a global solutions to a semi-linear wave equation with initial data of non-compact support in low space dimensions, Hokkaido Math. J. 22 (1993), 123–180.

    MATH  MathSciNet  Google Scholar 

  72. K. Kubota and K. Mochizuki, On small data scattering for 2-dimensional semilinear wave equations, Hokkaido Math. J. 22 (1993), 79–97.

    MathSciNet  Google Scholar 

  73. K. Kubota and K. Yokoyama, Global existence of classical solutions to systems of nonlinear wave equations with different speeds of propagation, Japanese J. Math. 27 (2001), 113–202.

    MathSciNet  Google Scholar 

  74. Li Ta-tsien and Chen Yun-Mei, Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations 13 (1988), 383–422.

    MathSciNet  Google Scholar 

  75. Li Ta-tsien and Zhou Yi, Nonlinear stability for two space dimensional wave equations with higher order perturbations, Nonlinear World 1 (1994), 35–58.

    MathSciNet  Google Scholar 

  76. H. Lindblad, Blow-up for solutions of □u = |u|pwith small initial data, Comm. Partial Differential Equations 15 (1990), 757–821.

    MATH  MathSciNet  Google Scholar 

  77. H. Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math. 45 (1992), 1063–1096.

    MATH  MathSciNet  Google Scholar 

  78. H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), 375–426.

    Article  MathSciNet  Google Scholar 

  79. H. Lindblad and C. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math. 118 (1996), 1047–1135.

    MathSciNet  Google Scholar 

  80. A. Majda, “Compressible fluid flow and systems of conservation laws”, Appl. Math. Sci. 53, Springer-Verlag, 1984.

    Google Scholar 

  81. G. P. Menzala and Y. Ebihara, Large time behavior of solutions to nonlinear systems of Klein-Gordon equations, Math. Aplic. Comp. 6 (1987), 69–96.

    Google Scholar 

  82. K. Mochizuki and T. Motai, The scattering theory for the nonlinear wave equation with small data, J. Math. Kyoto Univ. 25 (1985), 703–790.

    MathSciNet  Google Scholar 

  83. K. Mochizuki and T. Motai, The scattering theory for the nonlinear wave equation with small data, II, Publ. RIMS, Kyoto Univ. 23 (1985), 771–790.

    MathSciNet  Google Scholar 

  84. M. Nakamura and T. Ozawa, Small solutions to nonlinear wave equations in Sobolev spaces, Houston J. Math. 27 (2001), 613–632.

    MathSciNet  Google Scholar 

  85. M. Ohta, Counterexample to global existence for systems of nonlinear wave equations with different propagation speeds, Funkcialaj Ekvacioj 46 (2003), 471–477.

    Article  MATH  MathSciNet  Google Scholar 

  86. T. Ozawa, K. Tsutaya and Y. Tsutsumi, Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions, Math. Ann. 313 (1999), 127–140.

    Article  MathSciNet  Google Scholar 

  87. H. Pecher, Scattering for semilinear wave equations with small data in three space dimensions, Math. Z. 198 (1988), 277–289.

    Article  MATH  MathSciNet  Google Scholar 

  88. H. Pecher, Self-similar and asymptotically self-similar solutions of nonlinear wave equations, Math. Ann. 316 (2000), 259–281.

    Article  MATH  MathSciNet  Google Scholar 

  89. H. Pecher, Sharp existence results for self-similar solutions of semilinear wave equations, Nonlinear Differ. Equ. Appl. 7 (2000), 323–341.

    MATH  MathSciNet  Google Scholar 

  90. M. A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations 12 (1987), 677–700.

    MATH  MathSciNet  Google Scholar 

  91. F. Ribaud and A. Youssfi, Solutions globales et solutions auto-similaires de l’équation des ondes non linéaire, C. R. Acad. Sci. Paris, Série 1 329 (1999), 33–36.

    MathSciNet  Google Scholar 

  92. J. Schaeffer, The equation utt − △u = |u|pfor the critical value of p, Proc. Roy. Soc. Edinburgh 101A (1985), 31–44.

    MathSciNet  Google Scholar 

  93. J. Schaeffer, Finite time blow-up for utt − △u = H(ur, ut) in two space dimensions, Comm. Partial Differential Equations 11 (1986), 513–543.

    MATH  MathSciNet  Google Scholar 

  94. T. C. Sideris, Global behavior of solutions to nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations 8 (1983), 1291–1323.

    MATH  MathSciNet  Google Scholar 

  95. T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations 52 (1984), 378–406.

    Article  MATH  MathSciNet  Google Scholar 

  96. T. C. Sideris, The null condition and global existence of nonlinear elastic waves, Invent. Math. 123 (1996), 323–342.

    MATH  MathSciNet  Google Scholar 

  97. T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. 151 (2000), 849–874.

    MATH  MathSciNet  Google Scholar 

  98. T. C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal. 33 (2001), 477–488.

    Article  MathSciNet  Google Scholar 

  99. C.D. Sogge, Global existence for nonlinear wave equations with multiple speeds, Harmonic Analysis at Mount Holyoke, Contemp. Math. No. 320, Amer. Math. Soc., Providence, RI, 2003.

    Google Scholar 

  100. W. A. Strauss, Decay and asymptotics for □u = F(u), J. Funct. Anal. 2 (1968), 409–457.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  102. W. A. Strauss, “Nonlinear wave equations”, CBMS Regional Conference Series in Mathematics, 73, American Math. Soc., Providence, RI, 1989.

    Google Scholar 

  103. H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. Differential Equations 192 (2003), 308–325.

    Article  MATH  MathSciNet  Google Scholar 

  104. H. Sunagawa, A note on the large time asymptotics for a system of Klein-Gordon equations, Hokkaido Math. J. 33 (2004), 457–472.

    MATH  MathSciNet  Google Scholar 

  105. H. Sunagawa, Large time asymptotics of solutions to nonlinear Klein-Gordon systems, to appear in: Osaka J. Math.

    Google Scholar 

  106. H. Takamura, Global existence for nonlinear wave equations with small data of noncompact support in three space dimensions, Comm. Partial Differential Equations 17 (1992), 189–204.

    MATH  MathSciNet  Google Scholar 

  107. H. Takamura, An elementary proof of the exponential blow-up for semilinear wave equations, Math. Meth. Appl. Sci. 17 (1994), 239–249.

    Article  MATH  MathSciNet  Google Scholar 

  108. H. Takamura, Blow-up for semilinear wave equations with slowly decaying data in high dimensions, Differential Integral Equations 8 (1995), 647–661.

    MATH  MathSciNet  Google Scholar 

  109. D. Tataru, Strichartz estimates in the hyperbolic space and global existence for semilinear wave equation, Trans. Amer. Math. Soc. 353 (2000), 795–807.

    MathSciNet  Google Scholar 

  110. K. Tsutaya, Global existence theorem for semilinear wave equations with noncompact data in two space dimensions, J. Differential Equations 104 (1993), 332–360.

    Article  MATH  MathSciNet  Google Scholar 

  111. K. Tsutaya, Scattering theory for semilinear wave equations with small data in two space dimensions, Trans. A.M.S. 342 (1994), 595–618.

    MATH  MathSciNet  Google Scholar 

  112. K. Tsutaya, Global existence and the life span of solutions of semilinear wave equations with data of noncompact support in three space dimensions, Funkcial. Ekvac. 37 (1994), 1–18.

    MATH  MathSciNet  Google Scholar 

  113. N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equation with small data, Tsukuba J. Math. 22 (1998), 193–211.

    MATH  MathSciNet  Google Scholar 

  114. K. Yokoyama, Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan 52 (2000), 609–632.

    MATH  MathSciNet  Google Scholar 

  115. B. Yordanov and Q. Zhang, Finite time blow-up for critical wave equations in high dimensions, Preprint.

    Google Scholar 

  116. Y. Zhou, Blow-up of classical solutions to □u = |u|1+α in three space dimensions, J. Partial Differential Equations 5 (1992), 21–32.

    MATH  MathSciNet  Google Scholar 

  117. Y. Zhou, Life span of classical solutions to □u = |u|pin two space dimensions, Chinese Ann. Math. 14B (1993), 225–236.

    Google Scholar 

  118. Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Differential Equations 8 (1995), 135–144.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, Osaka Toyonaka, 560-0043, Japan

    Hideo Kubo

  2. Department of Mathematics, Faculty of Science, Saitama University, Saitama, 338-8570, Japan

    Masahito Ohta

Authors
  1. Hideo Kubo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masahito Ohta
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fakultät für Mathematik und Informatik Institut für Angewandte Analysis, TU Bergakademie Freiberg, 09596, Freiberg, Germany

    Michael Reissig

  2. Institut für Mathematik, Universität Potsdam, 14415, Potsdam, Germany

    Bert-Wolfgang Schulze

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Kubo, H., Ohta, M. (2005). On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations. In: Reissig, M., Schulze, BW. (eds) New Trends in the Theory of Hyperbolic Equations. Operator Theory: Advances and Applications, vol 159. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7386-5_2

Download citation

Publish with us

Policies and ethics

Search

Navigation