On the decay of solutions to a class of defocusing NLS

We consider the following family of Cauchy problems:\begin{equation*}i\partial_t u= \Delta u - u|u|^\alpha, (t,x) \in \mathbb{R} \times \mathbb{R}^d\end{equation*}$$u(0)=\varphi\in H^1(\mathbb{R}^d)$$where $0<\alpha<\frac 4{d-2}$ for $d\geq 3$ and $0<\alpha<\infty$ for $d=1,2$. We prove that the $L^r$-norms of the solutions decay as $t\rightarrow \pm \infty$, 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 in [5] for $d\geq 3$ and in [8] for $d=1,2$, where the same decay results are proved under the extra assumption $\alpha >\frac 4d$.

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Published 1 January 2009