Small data scattering for nonlinear Schrödinger wave and Klein-Gordon equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 2, pp. 435-460.

Small data scattering for nonlinear Schrödinger equations (NLS), nonlinear wave equations (NLW), nonlinear Klein-Gordon equations (NLKG) with power type nonlinearities is studied in the scheme of Sobolev spaces on the whole space ${ℝ}^{n}$ with order $s. The assumptions on the nonlinearities are described in terms of power behavior ${p}_{1}$ at zero and ${p}_{2}$ at infinity such as $1+4/n\le {p}_{1}\le {p}_{2}\le 1+4/\left(n-2s\right)$ for NLS and NLKG, and $1+4/\left(n-1\right)\le {p}_{1}\le {p}_{2}\le 1+4/\left(n-2s\right)$ for NLW.

Classification : 35L70
@article{ASNSP_2002_5_1_2_435_0,
author = {Nakamura, Makoto and Ozawa, Tohru},
title = {Small data scattering for nonlinear Schr\"odinger wave and Klein-Gordon equations},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {435--460},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 1},
number = {2},
year = {2002},
zbl = {1121.35086},
mrnumber = {1991146},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2002_5_1_2_435_0/}
}
Nakamura, Makoto; Ozawa, Tohru. Small data scattering for nonlinear Schrödinger wave and Klein-Gordon equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 2, pp. 435-460. http://www.numdam.org/item/ASNSP_2002_5_1_2_435_0/

[1] J. Bergh - J. Löfström, “Interpolation Spaces”, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin-New York, 1976. | Zbl 0344.46071

[2] T. Cazenave, “An Introduction to Nonlinear Schrödinger Equations”, Textos de Métodos Matemáticos 22, Instituto de Matemática, Rio de Janeiro, 1989.

[3] T. Cazenave - A. Haraux, Équation d'évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse 2 (1980), 21-51. | Numdam | MR 583902 | Zbl 0411.35051

[4] T. Cazenave - F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in ${H}^{s}$, Nonlinear Anal. 14 (1990), 807-836. | MR 1055532 | Zbl 0706.35127

[5] J. Ginibre - G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), 50-68. | MR 1351643 | Zbl 0849.35064

[6] J. Ginibre - G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z. 189 (1985), 487-505. | MR 786279 | Zbl 0549.35108

[7] T. Kato, On nonlinear Schrödinger equations 67 (1995), 281-306. | MR 1383498 | Zbl 0848.35124

[8] T. Kato, Nonlinear Schrödinger equations, In: “Schrödinger Operators”, H. Holden - A. Jensen (eds.), Lecture Notes in Phys. 345, Springer, Berlin, 1989, pp. 218-263. | MR 1037322 | Zbl 0698.35131

[9] M. Keel - T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980. | MR 1646048 | Zbl 0922.35028

[10] H. Lindblad - C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), 357-426. | MR 1335386 | Zbl 0846.35085

[11] S. Machihara - K. Nakanishi - T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations, preprint. | MR 1895710 | Zbl 0991.35080

[12] M. Nakamura - T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces, Publ. Res. Inst. Math. Sci. 37 (2001), 255-293. | MR 1855424 | Zbl 1006.35068

[13] M. Nakamura - T. Ozawa, The Cauchy problem for nonlinear wave equations in the homogeneous Sobolev space, Ann. Inst. H. Poincaré, Physique théorique 71 (1999), 199-215. | Numdam | MR 1705131 | Zbl 0960.35066

[14] M. Nakamura - T. Ozawa, Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces, Reviews in Math. Phys. 9 (1997), 397-410. | MR 1446653 | Zbl 0876.35080

[15] H. Pecher, Solutions of semilinear Schrödinger equations in ${H}^{s}$, Ann. Inst. H. Poincaré Physique théorique 67 (1997), 259-296. | Numdam | MR 1472820 | Zbl 0888.35101

[16] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), 261-270. | MR 731347 | Zbl 0538.35063

[17] P. Ramond, “Field Theory”, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990. | MR 1083767 | Zbl 0984.81500

[18] W. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1981), 110-133. | MR 614228 | Zbl 0466.47006

[19] W. Strauss, Nonlinear scattering theory at low energy: sequel, J. Funct. Anal. 43 (1981), 281-293. | MR 636702 | Zbl 0494.35068

[20] H. Triebel, “Theory of Function Spaces”, Birkhäuser, Basel, 1983. | MR 781540 | Zbl 0546.46027

[21] M. Tsutsumi, Scattering of solutions of nonlinear Klein-Gordon equations in three space dimensions, J. Math. Soc. Japan 35 (1983), 521-538. | MR 702775 | Zbl 0554.35095

[22] B. Wang, Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in ${H}^{s}$, Discrete Contin. Dynam. Systems 5 (1999), 753-763. | MR 1722368 | Zbl 0973.35132