Nonlinear small data scattering for the wave equation in R4+1

Journal of the Mathematical Society of Japan

Volumn 50, Issue 2, 1998, Pages 253-292

  Hidano, Kunio ^a  

^a TOKYO METROPOLITAN UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000896291     PISSN: 00255645     EISSN: 18811167     Source Type: Journal    
DOI: 10.2969/jmsj/05020253     Document Type: Article

References (22)

1. J. Ginibre and G. Velo, Conformal invariance and time decay for non linear wave equations. I, Ann. Inst. H. Poincare Phys. Theor., 47 (1987), 221–261.
2. J. Ginibre and G. Veto, Scattering theory in the energy space for a class of non-linear wave equations, Comm. Math. Phys., 123 (1989), 535–573.
3. R.T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323–340.
4. K. Hidano and K. Tsutaya, Global existence and asymptotic behavior of solutions for nonlinear wave equations, Indiana Univ. Math. J., 44 (1995), 1273–1305.
5. L. Hormander, On Sobolev spaces associated with some Lie algebras, in Current Topics in Partial Differential Equations, Kinokuniya, Japan, 261–287, 1986.
6. F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235–268.
7. S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equations, Comm. Pure Appl. Math., 38 (1985), 321–332.
8. S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space Rn+1, Comm. Pure Appl. Math., 40 (1987), 111–117.
9. H. Kubo and K. Kubota, Asymptotic behaviors of radially symmetric solutions of Qn = |u|pfor super critical values p in odd space dimensions, Hokkaido Math. J., 24 (1995), 287–336.
10. K. Kubota and K. Mochizuki, On small data scattering for 2-dimensional semilinear wave equations, Hokkaido Math. J., 22 (1993), 79–97.
11. T.T. Li and X. Yu, Life-span of classical solutions to fully nonlinear wave equations, Comm. Partial Differential Equations, 16 (1991), 909–940.
12. H. Lindblad and C.D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357–426.
13. K. Mochizuki and T. Motai, The scattering theory for the nonlinear wave equation with small data I. J. Math. Kyoto Univ. 25 (1985), 703–715.
14. K. Mochizuki and T. Motai, The scattering theory for the nonlinear wave equation with small data II, Publ. Res. Inst. Math. Sci., 23 (1987), 771–790.
15. T. Ozawa and Y. Tsutsumi, The nonlinear Schrodinger limit and the initial layer of the Zakharov equations, Differential Integral Equations, 5 (1992), 721–745.
16. H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261–270.
17. H. Pecher, Scattering for semilinear wave equations with small data in three space dimensions, Math. Z., 198 (1988), 277–289.
18. J. Schaeffer, The equation utt - Au= |u|pfor the critical value of p, Proc. Roy. Soc. Edinburgh Sect. A, 101 (1985), 31–44.
19. T.C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378–406.
20. W.A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110–133.
21. K. Tsutaya, Scattering theory for semilinear wave equations with small data in two space dimensions, Trans. Amer. Math. Soc., 342 (1994), 598–618.
22. Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Partial Differential Equations, 8 (1995), 135–144.