Abstract We prove the finite time extinction property ( u ( t ) ≡ 0 on Ω for any t ⩾ T ⋆ , for some T ⋆ > 0 ) for solutions of the nonlinear Schrodinger problem i u t + Δ u + a | u | − ( 1 − m ) u = f ( t , x ) , on a bounded domain Ω of R N , N ⩽ 3 , a ∈ C with Im ( a ) > 0 (the damping case) and under the crucial assumptions 0 m 1 and the dominating condition 2 m Im ( a ) ⩾ ( 1 − m ) | Re ( a ) | . We use an energy method as well as several a priori estimates to prove the main conclusion. The presence of the non-Lipschitz nonlinear term in the equation introduces a lack of regularity of the solution requiring a study of the existence and uniqueness of solutions satisfying the equation in some different senses according to the regularity assumed on the data.

中文翻译：

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有界域中强阻尼非线性薛定谔方程的有限时间消光

摘要 我们证明了非线性薛定谔问题 iut + Δ u + a | 的解的有限时间消光性质 ( u ( t ) ≡ 0 on Ω for any t ⩾ T ⋆ , for some T ⋆ > 0 ) 你| − ( 1 − m ) u = f ( t , x ) ，在 RN 的有界域 Ω 上，N ⩽ 3 ，a ∈ C 且 Im ( a ) > 0（阻尼情况）并且在关键假设 0 m 1 下和支配条件 2 m Im ( a ) ⩾ ( 1 − m ) | 再 ( a ) | . 我们使用能量方法以及几个先验估计来证明主要结论。方程中非Lipschitz 非线性项的存在导致解缺乏规律性，需要根据对数据假定的规律性研究在某些不同意义上满足方程的解的存在性和唯一性。