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 $PW$ of $H^1({\mathbb{R}}^d)$ for $d\ge 1$, and investigate behavior of the solutions with initial data in this set. For this end, we divide $PW$ 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.

References in corpus (4)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Scattering for the non-radial 3D cubic nonlinear Schroedinger equation
• On blow-up solutions to the 3D cubic nonlinear Schroedinger equation

Cited by in corpus (14)

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• Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance
• Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on ${\mathbb{R}}^3$
• Scattering below the ground state for the 2D non-linear Schrödinger and Klein-Gordon equations revisited
• Scattering of radial data in the focusing NLS and generalized Hartree equations
• Dynamics of the focusing 3D cubic NLS with slowly decaying potential
• Nonscattering solutions to the $L^2$-supercritical NLS Equations
• Scattering for the cubic Klein--Gordon equation in two space dimensions
• Cubic-quintic NLS: scattering beyond the virial threshold
• Existence of a ground state and scattering for a nonlinear Schroedinger equation with critical growth
• Scattering theory for nls with inverse-square potential in 2d