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arXiv 0811.1849math.APmath-phmath.MP
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一类散焦NLS方程解的衰减性

On the decay of solutions to a class of defocusing NLS

Nicola Visciglia

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摘要：

我们考虑下面的柯西问题族：{方程 *} iPDMATHABMMu = Δ u-uPDMATHABMN ^α，（t，x）\in \R \times \R^d {方程 *} $$u(0)=φ\in H^1(\R^d)$$其中$0<α<\frac 4{d-2}$用于$d\geq 3$，$0<α<\infty$用于$d=1,2$。我们证明了$L^r$ -规范的解决方案$t\to \pm \infty$衰减，提供$2<r<\frac{2d}{d-2}$时$d\geq 3$和$2<r<\infty$时$d=1,2$。特别是，我们扩展了以前的结果由Ginibre和Velo的$d\geq 3$和Nakanishi的$d=1,2$，其中相同的衰变结果证明下额外的假设$α>\frac 4d$。

英文摘要：

We consider the following family of Cauchy problems: {equation*} i\partial_t u= Δu - u|u|^α, (t,x) \in \R \times \R^d {equation*} $$u(0)=φ\in H^1(\R^d)$$ where $0<α<\frac 4{d-2}$ for $d\geq 3$ and $0<α<\infty$ for $d=1,2$. We prove that the $L^r$-norms of the solutions decay as $t\to \pm \infty$, provided that $2<r<\frac{2d}{d-2}$ when $d\geq 3$ and $2<r<\infty$ when $d=1,2$. In particular we extend previous results obtained by Ginibre and Velo for $d\geq 3$ and by Nakanishi for $d=1,2$, where the same decay results are proved under the extra assumption $α>\frac 4d$.

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