On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

Abstract

Abstract. We consider the Cauchy problem for the nonlinear Schrödinger equations

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

“…They split φ into the sum of a large L 2 function φ N and a small remainder function ψ N to obtain global solution of (C). In [6,7], we considered nonlinear Schrödinger equations with pure power nonlinearity |u| α−1 u in general space dimensions and obtained similar global existence results for a class of data which is not included in L 2 , and as a corollary we constructed global solutions for L p data p < 2 when α < α 0 where α 0 < 1 + 4/n is some power. One aim of the present paper is to establish a global solution of the Cauchy problem (H) with any mass subcritical nonlinearity (i.e.…”

“…We consider two solutions of (H). 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).…”

“…Basically, we follow the arguments in [6,7,10], which prove the global existence of the Cauchy problem of Schrödinger equations with power type nonlinearities |u| α−1 u for data with an infinite L 2 norm. In [6,10], the standard Strichartz inequalities (2.2) were used to estimate the nonlinear term and in [7], the solutions were established in function spaces based on admissible pairs but these approaches are not suitable for L p -framework. In fact, concerning the case of φ ∈ L p , the global existence results in these previous studies cannot cover all the mass-subcritical nonlinearities |u| α−1 u, 1 < α < 1 + 4/n.…”

“…In fact, concerning the case of φ ∈ L p , the global existence results in these previous studies cannot cover all the mass-subcritical nonlinearities |u| α−1 u, 1 < α < 1 + 4/n. For instance, the global result of [7] for L p -data is obtained only for 1 < α < 1 + 3/n while global solutions are expected to exist for any α ∈ (1, 1 + 4/n). To overcome this difficulty, in this paper, we establish solutions in different spaces L q (L r ) based on the generalized Strichartz estimate (2.5), which is more suitable for L p framework.…”

Well-posedness and decay estimates for 1D nonlinear Schrödinger equations with Cauchy data in $L^p$

“…Nonetheless, in [20], Vargas and Vega studied the 1D cubic case and by means of a splitting argument, they showed that if φ / ∈ L 2 but φ is sufficiently close to an L 2 -function in some sense, then a global solution can be established. Later in [14], [15], [16] we considered the case of more general power α and use their approach to construct a global solution for L p -Cauchy data if p < 2 is sufficiently close to 2. This is based on the natural idea that any L p -function is "close" to an L 2 -function if p is near 2.…”