We consider the Cauchy problem for the Schrödinger equation with a quadratic nonlinearity in one space dimensio iut + 1/2uxx = t-a\ux|2, u(0,x) = U0(X), where a G (0, 1). From the heuristic point of view, solutions to this problem should have a quasilinear character when o. G (1/2, 1). We show in this paper that the solutions do not have a quasilinear character for all a G (0, 1) due to the special structure of the nonlinear term. We also prove that for a G [1/2, 1) if the initial data UQ G H ' H H ' are small, then the solution has a slow time decay such as t-a/2 . For o. G (0, 1/2), if we assume that the initial data U0 are analytic and small, then the same time decay occurs.
|Journal||Electronic Journal of Differential Equations|
|Publication status||Published - 2001 Dec 1|
- Large time behaviour
- Quadratic nonlinearity
- Schrödinger equation
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