Rapidly decaying solutions of the nonlinear Schrödinger equation. (English) Zbl 0763.35085

The authors consider the global solutions in \(\mathbb{R}^ N\) of the nonlinear Schrödinger equation (NSE), \[ iu_ t+\Delta u=\lambda| u|^ \alpha u \] with \(\lambda\in\mathbb{R}\), \(0<\alpha<4/(N-2)\). For \(\lambda>0\), \(\alpha\geq\alpha_ 0=(2-N+\sqrt{N^ 2+12N+4})/(2N)\), or \(\lambda<0\), \(\alpha>4/(N+2)\), the existence of a scattering theory in \[ X=\{u\in H^ 1(R^ N);\;| x| u\in L^ 2(R^ N)\} \] is obtained. For \(\alpha\geq\alpha_ 0\) and \(\varphi\in X\), the authors also prove that the solution of (NSE) with initial data \(\varphi(x)\exp\{ib| x|^ 2/4\}\) is rapidly decaying as \(t\) goes to infinity, here \(b\) is sufficiently large. Also, the authors extend the lower bound on \(\alpha\) to \(4/(N+2)\), namely \(4/(N+2)<\alpha<4/(N-2)\), for which a low energy scattering theory of (NSE) in \(X\) exists.
The main result of this paper concerning scattering theory of (NSE) generalizes some work of Y. Tsutsumi [Ann. Inst. Henri Poincaré, Phys. Théor. 43, 321-347 (1985; Zbl 0612.35104)] and N. Hayashi and Y. Tsutsumi [Lect. Notes Math. 1285, 162-168 (1987; Zbl 0633.35059)], who proved similar results for \(\lambda>0\), \(\alpha>\alpha_ 0\).


35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI


[1] Barab, J. E.: Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation. J. Math. Phys.25, 3270–3273 (1984) · Zbl 0554.35123 · doi:10.1063/1.526074
[2] Berestycki, H., Gallouet, T., Kavian, O.: Équations de champs scalaires Euclidiens nonlinéaires dans le plan. C. Rend. Acad. Sci. Paris297, 307–310 (1983) · Zbl 0544.35042
[3] Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. Arch. Rat. Mech. Anal.82, 313–375 (1983) · Zbl 0556.35046
[4] Cazenave, T.: An introduction to nonlinear Schrödinger equations. Textos de Métodos Matemáticos vol.22. I.M.U.F.R.J., Rio de Janeiro 1989
[5] Cazenave, T., Weissler, F. B.: The Cauchy problem for the nonlinear Schrödinger equation inH 1. Manuscripta Math.61, 477–494 (1988) · Zbl 0696.35153 · doi:10.1007/BF01258601
[6] Cazenave, T., Weissler, F. B.: The Cauchy problem for the critical nonlinear Schrödinger equation inH s. Nonlinear Anal, TMA14, 807–836 (1990) · Zbl 0706.35127 · doi:10.1016/0362-546X(90)90023-A
[7] Cazenave, T., Weissler, F. B.: The structure of solutions to the pseudo conformally invariant nonlinear Schrödinger equation. Proc. Royal Soc. Edinburgh117A, 251–273 (1991) · Zbl 0733.35094
[8] Constantin, P., Saut, J.-C.: Local smoothing properties of dispersive equations. J. Am. Math. Soc.1, 413–439 (1988) · Zbl 0667.35061 · doi:10.1090/S0894-0347-1988-0928265-0
[9] Constantin, P., Saut, J.-C.: Local smoothing properties of Schrödinger equations. Indiana Univ. Math. J.38, 791–810 (1989) · Zbl 0712.35022 · doi:10.1512/iumj.1989.38.38037
[10] Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. J. Funct. Anal.32, 1–71 (1979) · Zbl 0396.35028 · doi:10.1016/0022-1236(79)90076-4
[11] Ginibre, J., Velo, G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. Henri Poincaré, Analyse Non Linéaire2, 309–327 (1985) · Zbl 0586.35042
[12] Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pure Appl.64, 363–401 (1985) · Zbl 0535.35069
[13] Ginibre, J., Velo, G.: Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations. Ann. Inst. Henri Poincaré, Physique Théorique43, 399–442 (1985) · Zbl 0595.35089
[14] Glassey, R. T.: On the asymptotic behavior of nonlinear wave equations. Trans. Am. Math. Soc.182, 187–200 (1973) · Zbl 0269.35009 · doi:10.1090/S0002-9947-1973-0330782-7
[15] Glassey, R. T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys.18, 1794–1797 (1977) · Zbl 0372.35009 · doi:10.1063/1.523491
[16] Hayashi, N., Ozawa, T.: Lower bounds for order of decay or of growth in time for solutions to linear and nonlinear Schrödinger equations. Publ. Res. Inst. Math. Sci. (Kyoto Univ.) vol.560. Kyoto Univ, 1986 · Zbl 0714.35014
[17] Hayashi, N., Tsutsumi, Y.: Remarks on the scattering problem for nonlinear Schrödinger equations. In: Differential Equations and Mathematical Physics. Lecture Notes in Math. vol.1285, pp. 162–168 Berlin, Heidelberg, New York: Springer 1987 · Zbl 0633.35059
[18] Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Physique Théorique46, 113–129 (1987)
[19] Keing, C., Ponce, G., Vega, L.: Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J.40, 33–69 (1991) · Zbl 0738.35022 · doi:10.1512/iumj.1991.40.40003
[20] Stein, E. M.: Singular integrals and differentiability of functions. Princeton, NJ: Princeton University Press, 1970 · Zbl 0207.13501
[21] Strauss, W. A.: Nonlinear scattering theory. In: Scattering Theory in Mathematical Physics. pp. 53–78 (eds.). Lavita, J. A., Planchard, J.-P., Dordrecht: Reidel 1974
[22] Strauss, W. A.: Nonlinear scattering theory at low energy. J. Funct. Anal.41, 110–133 (1981) · Zbl 0466.47006 · doi:10.1016/0022-1236(81)90063-X
[23] Strauss, W. A.: Nonlinear wave equations. Regional Conference Series in Mathematics vol.73, Providence, RI: Am. Math. Soc., 1989 · Zbl 0714.35003
[24] Strauss, W. A.: The nonlinear Schrödinger equation. In: Contemporary developments in continuum mechanics and partial differential equations, pp. 452–465. De La Penha and Medeiros (eds.). Amsterdam: North-Holland, 1978
[25] Strichartz, M.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J.44, 705–714 (1977) · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1
[26] Tsutsumi, Y.: Scattering problem for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Physique Théorique43, 321–347 (1985) · Zbl 0612.35104
[27] Tsutsumi, Y., Yajima, K.: The asymptotic behavior of nonlinear Schrödinger equations. Bull. Am. Math. Soc.11, 186–188 (1984) · Zbl 0555.35028 · doi:10.1090/S0273-0979-1984-15263-7
[28] Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys.110, 415–426 (1987) · Zbl 0638.35036 · doi:10.1007/BF01212420
[29] Yajima, K.: The surfboard Schrödinger equations. Commun. Math. Phys.96, 349–360 (1984) · Zbl 0599.35037 · doi:10.1007/BF01214580
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.