Large data mass-subcritical NLS: critical weighted bounds imply scattering

Abstract

Abstract. We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity sc ∈ (max{−1, − d 2 }, 0), we prove that any solution satisfyingx < ∞ on its maximal interval of existence must be global and scatter.

The radial mass-subcritical NLS in negative order Sobolev spaces

“…Scattering in weighted spaces. As mentioned in the introduction, Theorem 1.1 yields an improvement to our previous results in [20] for a range of nonlinearities in the radial defocusing case. Specifically, in certain cases we can show that solutions with radial data in the weighted space…”

“…The case t → −∞ is similar. To this end, we recall the operator J s (t) = e it∆ |x| s e −it∆ = e i|x| 2 /4t (−4t 2 ∆) s/2 e −i|x| 2 /4t , which featured prominently in [20]. We will first show that J |sc| u remains bounded in L 2 .…”

“…We would like to point out that when compared with Theorem 1.1, the result in [20] has strictly stronger hypotheses and conclusions. Indeed, by unitarity of e it∆ , Sobolev embedding, and Hölder's inequality, one has |∇| sc u(t) L 2 x e −it∆ u(t) L dp 2…”

“…We will first show that J |sc| u remains bounded in L 2 . To see this, we use the Strichartz estimates in [20] (see e.g. Proposition 2.6 therein) and the Duhamel formula…”

Global dynamics of solutions with group invariance for the nonlinear Schrödinger equation

“…For other studies of global dynamics of dispersive equations, see [16,42,17] (other global dynamics of (NLS)), [25,15,33,30,9,41] (energy-critical NLS), [31,29,10] (mass-critical NLS), [39,40,27] (mass-subcritical NLS), [26,14,13] (wave equations), [23,28,43,24] (nonlinear Klein-Gordon equations), and references therein.…”