Analytic Smoothing Effect for the Nonlinear Schrödinger Equations Without Square Integrability

Global solutions to the Hartree equation for large L^{p}-initial data

“…Although his work [11] relies on a subtle cancellation of the cubic nonlinearity uuū and his approach cannot be applied to the pure power case |u| α−1 u, it is plausible that properties similar to (1.2) hold for solutions of other nonlinear dispersive equations. In fact, in [4,5], we showed that local solutions of (H) for L p data satisfying (1.2) can be established and this is the second aim in the present paper. Roughly speaking, the goal of the present paper is as follows: construct a global solution of (H) for large data φ ∈ L p satisfying the property (1.2) at least on a finite time interval [−T − , T + ].…”

“…One is a global solution which is constructed by means of the splitting argument similar to the ones in [6,7,10] but we establish the solution in different function spaces based on the generalized Strichartz estimates, which is more suitable in the L p -framework. The other is a local solution based on [4,5] which is defined on some finite interval I and satisfies (1.2). Then we get a desired global solution if we showed that these two solutions are identical on some finite interval.…”