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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
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