Continuous Dependence of Cauchy Problem For Nonlinear Schrödinger Equation in $H^s$

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u={\lambda }_{0}u+{\lambda }_1|u{|}^{\alpha }u$ in ${\mathbb{R}}^N$, where ${\lambda }_0,{\lambda }_1\in \mathbb{C}$, in $H^s$ subcritical and critical case: $0<\alpha \le \frac{4}{N-2s}$ when $1<s<\frac{N}{2}$ and $0<\alpha <+\infty$ when $s\ge \frac{N}{2}$. We show that the solution depends continuously on the initial value in the standard sense in $H^s({\mathbb{R}}^N)$ if $\alpha$ satisfies certain assumptions.

48pages, no figure. arXiv admin note: text overlap with arXiv:1006.2745 by other authors

References in corpus (1)

• Continuous dependence for NLS in fractional order spaces

Cited by in corpus (3)

• On the unconditional uniqueness for NLS in $H^s$
• Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation
• Continuous dependence for $H^2$ critical nonlinear Schrödinger equations in high dimensions