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On solutions on the initial value problem for the nonlinear Schrödinger equations. (English) Zbl 0657.35033

Consider the initial value problem for the nonlinear Schrödinger (NLS) equation of the form \[ (1.1)\quad iu_ t+\Delta u=f(| u|^ 2)u,\quad (t,x)\in {\mathbb{R}}\times {\mathbb{R}}^ n, \]
\[ (1.2)\quad u(0,x)=\phi (x),\quad x\in {\mathbb{R}}^ n, \] where \(\Delta\) denotes the n- dimensional Laplacian. The NLS equation (1.1) appears in various physical applications, such as plasma physics, nonlinear optics, and nonrelativistic quantum physics. Recently Y. Tsutsumi [Funkc. Ekvacioj, Ser. Int. 30, 115-125 (1987; Zbl 0638.35021)] has obtained the global existence of solutions in a weaker class than in \(H^ 1({\mathbb{R}}^ n)\), under the hypotheses, \(\phi \in L^ 2\) and \(f(s)=s^{(p-1)/2}\) with \(1<p<1+(4/n)\). There is, however, no result concerning the regularizing effect of the NLS equation, analogous to that of the (nonlinear) heat equation or to that of the KdV equation which is obtained by T. Kato [Res. Notes Math. 53, 293-307 (1981; Zbl 0542.35064) and Adv. Math., Suppl. Stud. 8, 93-128 (1983; Zbl 0549.34001)]. The question considered here is whether or not solutions of (1.1), (1.2) become smooth for \(t\neq 0\) even if initial functions are not smooth.
Our purpose in this paper is to show that if the initial function \(\phi\) decreases sufficiently rapidly, then the solution of (1.1), (1.2) becomes smooth for \(t\neq 0\), provided the nonlinear term \(f(| u|^ 2)\) is smooth.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35B65 Smoothness and regularity of solutions to PDEs
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