## 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 $$\lambda>0$$, $$\alpha>\alpha_ 0$$.

### MSC:

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

### Keywords:

global solutions; existence of a scattering theory

### Citations:

Zbl 0612.35104; Zbl 0633.35059
Full Text:

### References:

 [1] Barab, J. E.: Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation. J. Math. Phys.25, 3270–3273 (1984) · Zbl 0554.35123 [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 [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 [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 [9] Constantin, P., Saut, J.-C.: Local smoothing properties of Schrödinger equations. Indiana Univ. Math. J.38, 791–810 (1989) · Zbl 0712.35022 [10] Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. J. Funct. Anal.32, 1–71 (1979) · Zbl 0396.35028 [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 [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 [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 [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 [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 [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 [28] Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys.110, 415–426 (1987) · Zbl 0638.35036 [29] Yajima, K.: The surfboard Schrödinger equations. Commun. Math. Phys.96, 349–360 (1984) · Zbl 0599.35037
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