Small data scattering for nonlinear Schrödinger wave and Klein-Gordon equations

Nakamura, Makoto; Ozawa, Tohru

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 $s<n/2$. The assumptions on the nonlinearities are described in terms of power behavior $p_1$ at zero and $p_2$ at infinity such as $1+4/n\le p_1\le p_2\le 1+4/(n-2s)$ for NLS and NLKG, and $1+4/(n-1)\le p_1\le p_2\le 1+4/(n-2s)$ for NLW.

Classification : 35L70

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

[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. | Zbl | MR | Numdam

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

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

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

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

[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. | Zbl | MR

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

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

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

[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. | Zbl | MR

[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. | Zbl | MR | Numdam

[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. | Zbl | MR

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

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

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

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

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

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

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

[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. | Zbl | MR