On the sharp scattering threshold for the mass-energy double critical nonlinear Schrödinger equation via double track profile decomposition
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Contributors
Abstract
The present paper is concerned with the Iarge data scattering problem for the massenergy double critical nonlinear Schrodinger equation i[Formula presented] in H1(ℝd) with d ≥ 3, referred to as DCNLS. In the defocusing-defocusing regime, Tao, Visan and Zhang showed that the unique solution of DCNLS is global and scattering in time for arbitrary initial data in H1(ℝd). This does not hold when at least one of the nonlinearities is focusing, due to the possible formation of blow-up and soliton solutions. However, precise thresholds for a solution of DCNLS being scattering were open in all the remaining regimes. FoIIowing the classical concen-tration compactness principle, we impose sharp scattering thresholds in terms of ground states for DCNLS in all the remaining regimes. The new challenge arises from the fact that the remainders of the standard L2- or H1-profile decomposition fail to have asymptotically vanishing diagonal L2-and H1-Strichartz norms simultaneously. To overcome this difficulty, we construct a double track profile decomposition which is capable of capturing the low-, medium- and high-frequency bubbles within a single profile decomposition and possesses remainders that are asymptotically small in both of the diagonal L2- and H1 -Strichartz spaces.
Details
Original language | English |
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Pages (from-to) | 187-255 |
Number of pages | 69 |
Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |
Volume | 41 |
Issue number | 1 |
Publication status | Published - 2024 |
Peer-reviewed | Yes |
Keywords
ASJC Scopus subject areas
Keywords
- combined nonlinearities, Double critical Schrödinger equation, scattering threshold