On the decay of solutions to a class of defocusing NLS
Abstract
Description
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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