Blowup and scattering problems for the nonlinear Schrödinger equations

Abstract

We consider L 2 -supercritical and H 1 -subcritical focusing nonlinear Schrödinger equations. We introduce a subset P W of H 1 (R d ) for d ≥ 1, and investigate behavior of the solutions with initial data in this set. For this end, we divide P W into two disjoint components P W + and P W − . Then, it turns out that any solution starting from a datum in P W + behaves asymptotically free, and solution starting from a datum in P W − blows up or grows up, from which we find that the ground state has two unstable directions. We also investigate some properties of generic global and blowup solutions.It is worth while noting the following facts: 1. P W + and P W − are unbounded open sets in H 1 (R d ): Indeed, one can easily verify this fact by considering the scaled functions f λ (x) := λ 2. P W = P W + ∪ P W − and P W + ∩ P W − = ∅ (see Lemma 2.1) 3. P W + and P W − are invariant under the flow defined by the equation (1.1) (see Proposition 2.5 and Proposition 2.6).

Scattering below the ground state for the 2$d$ radial nonlinear Schrödinger equation

“…Observing that 1 1+σ = 1 1+min{2, p 2 } = max{ 1 3 , 2 p+2 } = α, we complete the proof of the proposition.…”

“…Theorem 1.1 result was previously established in [1,3,8], who extended the arguments of [6,9]. In fact, in [1,3,8] the same result is proven without the restriction to radial initial data. In these works the authors proceed via the concentrationcompactness approach to induction on energy.…”

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“…p C 2 D 2 ? in (1-2)); see [Akahori and Nawa 2010;Côte et al 2008;Duyckaerts et al 2008;Killip et al 2008;Krieger and Schlag 2009;Sterbenz and Tataru 2010;Tao 2008a;2008b;2008c;2009a;2009b].…”

“…It is further shared with the solution space (either the energy space or L 2 , i.e. the critical case), except for the NLS with a subcritical power [Duyckaerts et al 2008;Akahori and Nawa 2010]. The scaling invariance brings significant difficulties for the analysis, but also a lot of algebraic or geometric structures and simplifications.…”

“…The special feature of the critical wave Equation (1-6) is that those functionals are the same modulo constant multiples, which is exactly due to the scaling invariance. For the NLS with a subcritical power [Duyckaerts et al 2008;Akahori and Nawa 2010], the functionals are different from each other, but the situation is much better than NLKG, because they contain only two terms (without the L 2 norm), the L 2 is another conserved quantity, and the virial identity is used both for the blow-up and for the scattering, while K 1;0 is not so useful for NLS.…”