Global existence and scattering for the inhomogeneous nonlinear Schr\"odinger equation

TL;DR

This paper establishes global existence and scattering results for the inhomogeneous nonlinear Schrödinger equation with a potential, extending known ranges of the nonlinearity exponent and considering the effects of the potential's behavior at zero and infinity.

Contribution

The paper introduces new global existence and scattering results for the inhomogeneous NLS with a potential, extending the range of admissible nonlinearities and analyzing potential effects.

Findings

Global existence for oscillating initial data.

Scattering in weighted L^2 space for a new range of alpha.

Impact of potential behavior on solution properties.

Abstract

In this paper we consider the inhomogeneous nonlinear Schr\"odinger equation $i\partial_t u +\Delta u=K(x)|u|^\alpha u,\, u(0)=u_0\in H^s({\mathbb R}^N),\, s=0,\,1,$ $N\geq 1,$ $|K(x)|+|x|^s|\nabla^sK(x)|\lesssim |x|^{-b},$ $0<b<\min(2,N-2s),$ $0<\alpha<{(4-2b)/(N-2s)}$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range $\alpha_0(b)<\alpha<(4-2b)/N$. The value $\alpha_0(b)$ is the positive root of $N\alpha^2+(N-2+2b)\alpha-4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=\mu|x|^{-b}$, $\mu\in \mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $\alpha$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.