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In this paper, we prove the decay and scattering in the energy space for nonlinear Schrödinger equations with regular potentials in $\mathbb{R}^d$ namely, $i∂_tu+Δu-V(x)u+ λ|u|^{p-1}u=0$. We will prove decay estimate and scattering of the solution in the small data case when $1+\frac{2}{d}<p ≤ 1+\frac{4}{d-2}, d ≥ 3$. The index $1+\frac{2}{d}$ is sharp for scattering concerning the result of Strauss [22]. This result generalizes the one-dimensional work of Cuccagna et al. [4] to all $d ≥ 3$.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v50n3.17.05}, url = {http://global-sci.org/intro/article_detail/jms/10621.html} }In this paper, we prove the decay and scattering in the energy space for nonlinear Schrödinger equations with regular potentials in $\mathbb{R}^d$ namely, $i∂_tu+Δu-V(x)u+ λ|u|^{p-1}u=0$. We will prove decay estimate and scattering of the solution in the small data case when $1+\frac{2}{d}<p ≤ 1+\frac{4}{d-2}, d ≥ 3$. The index $1+\frac{2}{d}$ is sharp for scattering concerning the result of Strauss [22]. This result generalizes the one-dimensional work of Cuccagna et al. [4] to all $d ≥ 3$.