Scattering norm estimate near the threshold for the energy-subcritical NLS
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909947032240128 |
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| author | Ma, Zuyu |
| author_facet | Ma, Zuyu |
| contents | We consider the focusing energy-subcritical Schrödinger equations. In earlier works by Holmer-Roudenko \cite{holmer}, Duyckaerts-Holmer-Roudenko \cite{duyckaerts2}, Akahori-Nawa \cite{akahori}, Fang-Xie-Cazenave \cite{fang}, Guevara \cite{guevara} and later by Dodson-Murphy \cite{dodson1,dodson2} and Arora-Dodson-Murphy \cite{arora}, they proved that scattering is the only dynamical behavior if the $H^1$ initial data satisfies $M(u_0)^{(1-s_c)/s_c}E(u_0)<M(Q)^{(1-s_c)/s_c}E(Q)$ and $\| u\|^{(1-s_c)/s_c}_{L^2}\| u\|_{\dot{H}^1}<\| Q\|^{(1-s_c)/s_c}_{L^2}\|Q\|_{\dot{H}^1}$, where $Q$ is the ground state. In this paper, we establish asymptotic estimates for the upper bound of the scattering norms as $M(u_0)^{(1-s_c)/s_c}E(u_0)$ approaches the threshold mass-energy threshold $M(Q)^{(1-s_c)/s_c}E(Q)$, which generalizes the work of Duyckaerts-Merle \cite{duyckaerts} on the energy-critical Schrödinger equation($s_c=1$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scattering norm estimate near the threshold for the energy-subcritical NLS Ma, Zuyu Analysis of PDEs 35Q55 We consider the focusing energy-subcritical Schrödinger equations. In earlier works by Holmer-Roudenko \cite{holmer}, Duyckaerts-Holmer-Roudenko \cite{duyckaerts2}, Akahori-Nawa \cite{akahori}, Fang-Xie-Cazenave \cite{fang}, Guevara \cite{guevara} and later by Dodson-Murphy \cite{dodson1,dodson2} and Arora-Dodson-Murphy \cite{arora}, they proved that scattering is the only dynamical behavior if the $H^1$ initial data satisfies $M(u_0)^{(1-s_c)/s_c}E(u_0)<M(Q)^{(1-s_c)/s_c}E(Q)$ and $\| u\|^{(1-s_c)/s_c}_{L^2}\| u\|_{\dot{H}^1}<\| Q\|^{(1-s_c)/s_c}_{L^2}\|Q\|_{\dot{H}^1}$, where $Q$ is the ground state. In this paper, we establish asymptotic estimates for the upper bound of the scattering norms as $M(u_0)^{(1-s_c)/s_c}E(u_0)$ approaches the threshold mass-energy threshold $M(Q)^{(1-s_c)/s_c}E(Q)$, which generalizes the work of Duyckaerts-Merle \cite{duyckaerts} on the energy-critical Schrödinger equation($s_c=1$). |
| title | Scattering norm estimate near the threshold for the energy-subcritical NLS |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/2509.01505 |