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Semilinear hyperbolic systems violating the null condition

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Abstract

We consider systems of semilinear wave equations in three space dimensions with quadratic nonlinear terms not satisfying the null condition. We prove small data global existence of the classical solution under a new structural condition related to the weak null condition. For two-component systems satisfying this condition, we also observe a new kind of asymptotic behavior: Only one component is dissipated and the other one behaves like a non-trivial free solution in the large time.

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References

  1. Alinhac, S.: An example of blowup at infinity for quasilinear wave equations. Astérisque 284, 1–91 (2003)

    Google Scholar 

  2. Alinhac, S.: Semilinear hyperbolic systems with blowup at infinity. Indiana Univ. Math. J. 55, 1209–1232 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Hayashi, N., Li, C., Naumkin, P.I.: On a system of nonlinear Schrödinger equations in 2D. Differ. Integr. Equ. 24, 417–434 (2011)

    MATH  MathSciNet  Google Scholar 

  6. Hayashi, N., Li, C., Naumkin, P.I.: Modified wave operator for a system of nonlinear Schrödinger equations in 2d. Comm. Partial Differ. Equ. 37, 947–968 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hörmander, L.: \(L^1,\, L^\infty \) estimates for the wave operator. In: Analyse Mathématique et Applications, Contributions en l’Honneur de J. L. Lions, pp. 211–234. Gauthier-Villars, Paris (1988)

  8. Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations. Springer, Berlin (1997)

    MATH  Google Scholar 

  9. John, F.: Blow-up for quasilinear wave equations in three space dimensions. Comm. Pure Appl. Math. 34, 29–51 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  10. Katayama, S.: Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions. J. Hyperbolic Differ. Equ. 9, 263–323 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Katayama, S.: Asymptotic behavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. J. Differ. Equ. 255, 120–150 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Katayama, S., Kubo, H.: Asymptotic behavior of solutions to semilinear systems of wave equations. Indiana Univ. Math. J. 57, 377–400 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Katayama, S., Li, C., Sunagawa, H.: A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D. Differ. Integr. Equ. 27, 301–312 (2014)

    MathSciNet  Google Scholar 

  14. Katayama, S., Ozawa, T., Sunagawa, H.: A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions. Comm. Pure Appl. Math. 65, 1285–1302 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kawahara, Y., Sunagawa, H.: Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance. J. Differ. Equ. 251, 2549–2567 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Klainerman, S.: The null condition and global existence to nonlinear wave equations. Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1. Lectures in Applied Mathematics, vol. 23, pp. 293–326. American Mathematical Society, Providence (1986)

  17. Lax, P.D., Phillips, R.S.: Scattering Theory, Revised edn. Academic Press, New York (1989)

    MATH  Google Scholar 

  18. Li, C.: Decay of solutions for a system of nonlinear Schrödinger equations in 2D. Discrete Contin. Dyn. Syst. Ser. A 32, 4265–4285 (2012)

    Article  MATH  Google Scholar 

  19. Lindblad, H.: On the lifespan of solutions of nonlinear wave equations with small initial data. Comm. Pure Appl. Math. 43, 445–472 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Lindblad, H.: Global solutions of quasilinear wave equations. Amer. J. Math. 130, 115–157 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lindblad, H., Rodnianski, I.: The weak null condition for Einstein’s equations. C. R. Math. Acad. Sci. Paris 336, 901–906 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lindblad, H., Rodnianski, I.: Global existence for the Einstein vacuum equations in wave coordinates. Comm. Math. Phys. 256, 43–110 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lindblad, H., Rodnianski, I.: The global stability of Minkowski space-time in harmonic gauge. Ann. Math. 171, 1401–1477 (2010)

  25. Sogge, C.D.: Lectures on Non-Linear Wave Equations. International Press, Boston (1995)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to express their sincere gratitude to Prof. Akitaka Matsumura for his comments on the earlier version of this work. The work of S. K. is supported by Grant-in-Aid for Scientific Research (C) (No. 23540241), JSPS. The work of H. S. is supported by Grant-in-Aid for Young Scientists (B) (No. 22740089) and Grant-in-Aid for Scientific Research (C) (No. 25400161), JSPS.

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Authors and Affiliations

  1. Department of Mathematics, Wakayama University, 930 Sakaedani, Wakayama, 640-8510, Japan

    Soichiro Katayama

  2. Osaka Prefectural Tennoji High School, 2-4-23 Sanmeicho, Abeno-ku, Osaka, 545-0005, Japan

    Toshiaki Matoba

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

    Hideaki Sunagawa

Authors
  1. Soichiro Katayama
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  2. Toshiaki Matoba
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  3. Hideaki Sunagawa
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Correspondence to Soichiro Katayama.

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Dedicated to the memory of Professor Rentaro Agemi.

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Katayama, S., Matoba, T. & Sunagawa, H. Semilinear hyperbolic systems violating the null condition. Math. Ann. 361, 275–312 (2015). https://doi.org/10.1007/s00208-014-1071-1

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  • DOI: https://doi.org/10.1007/s00208-014-1071-1

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