Large-data $L^2$-decay for attractive-dissipative nonlinear Schrödinger equations without the strong dissipative condition
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914568931901440 |
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| author | Kita, Naoyasu Miyazaki, Hayato Sato, Takuya |
| author_facet | Kita, Naoyasu Miyazaki, Hayato Sato, Takuya |
| contents | We prove a large-data $L^2$-decay estimate for nonlinear dissipative Schrödinger equations with attractive-dissipative power nonlinearity. The main difficulty is the lack of sign definiteness of the standard energy when $\Reλ<0$, which prevents the usual energy argument from directly yielding a uniform gradient bound. We introduce an augmented energy, obtained by adding a suitable multiple of the decreasing $L^2$-norm to the standard energy. This produces an additional dissipative term and gives a direct uniform-in-time $H^1$ bound without the iteration argument used in previous works. Consequently, for arbitrary initial data in the weighted energy space $Σ= H^1 \cap \mathcal{F}H^1$, we obtain the decay rate previously known under the strong dissipative condition throughout the sharp decay range $1<p\le 1+2/d$. This removes the remaining restriction $p\le 1+4/(3d)$ in the attractive-dissipative case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15837 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large-data $L^2$-decay for attractive-dissipative nonlinear Schrödinger equations without the strong dissipative condition Kita, Naoyasu Miyazaki, Hayato Sato, Takuya Analysis of PDEs 35Q55, 35B40 We prove a large-data $L^2$-decay estimate for nonlinear dissipative Schrödinger equations with attractive-dissipative power nonlinearity. The main difficulty is the lack of sign definiteness of the standard energy when $\Reλ<0$, which prevents the usual energy argument from directly yielding a uniform gradient bound. We introduce an augmented energy, obtained by adding a suitable multiple of the decreasing $L^2$-norm to the standard energy. This produces an additional dissipative term and gives a direct uniform-in-time $H^1$ bound without the iteration argument used in previous works. Consequently, for arbitrary initial data in the weighted energy space $Σ= H^1 \cap \mathcal{F}H^1$, we obtain the decay rate previously known under the strong dissipative condition throughout the sharp decay range $1<p\le 1+2/d$. This removes the remaining restriction $p\le 1+4/(3d)$ in the attractive-dissipative case. |
| title | Large-data $L^2$-decay for attractive-dissipative nonlinear Schrödinger equations without the strong dissipative condition |
| topic | Analysis of PDEs 35Q55, 35B40 |
| url | https://arxiv.org/abs/2605.15837 |