Threshold scattering for the focusing NLS with a repulsive Dirac delta potential

Abstract

We establish the scattering of solutions to the focusing mass supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potentialwhere u 0 ∈ H 1 (R) is the initial data, Q is the ground state of the free NLS on the real line R, Eγ is the energy, M is the mass and σ = (p + 3)/(p − 5). We also prove failure of the uniform space-time bounds at the mass-energy threshold.

Threshold odd solutions to the nonlinear Schrödinger equation in one dimension

“…for all t. We get Proof of Proposition 2.13 (2). Suppose that u is global in positive time direction.…”

“…Proposition 6.3. We assume the same assumption in Proposition 2.13 (2). Suppose that u is global in positive time direction.…”

“…Corollary 6.4. We assume the assumption of Proposition 2.13 (2) and that the solution is global in positive time, then u blows up in negative time.…”

“…Idea of proof. Our proof of the scattering result is based on the contradiction argument by Duyckaerts, Landoulsi, and Roudenko [10], where they consider 3d cubic NLS outside an obstacle (see also [23] and [2]). If the statement fails, combining a modulation argument and a concentration compactness argument, we construct a critical element at the threshold which has the following compactness property: There exists x(t) such that u(t, •) − {ψ(t, • − x(t)) − ψ(t, − • −x(t))} converges to 0 for some ψ.…”

“…The main difficulty appears in the modulation argument. For NLS with potential as in [23,2], we consider the linearization around the ground state Q and thus we can control the translation parameter y easily. On the other hand, in our setting, we need to consider the linearization around Q(• − y) − Q(• + y).…”