On the decay of solutions to a class of defocusing NLS

Abstract

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*} $$u(0)=\phi \in H^1({\mathbb{R}}^d)$$ where $0<\alpha <\frac{4}{d-2}$ for $d\ge 3$ and $0<\alpha <\infty$ 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 $d\ge 3$ and $2<r<\infty$ when $d=1,2$. In particular we extend previous results obtained by Ginibre and Velo for $d\ge 3$ and by Nakanishi for $d=1,2$, where the same decay results are proved under the extra assumption $\alpha >\frac{4}{d}$.

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References in corpus (2)

• Bilinear virial identities and applications
• Tensor products and Correlation Estimates with applications to Nonlinear Schrödinger equations