论文标题
有限阻尼的非线性schr {Ö} dinger方程的有限时间灭绝
Finite time extinction for the strongly damped nonlinear Schr{ö}dinger equation in bounded domains
论文作者
论文摘要
我们证明\ textIt {有限的时间灭绝属性} $（u（t）\ equiv 0 $ on $ω$在任何$ t \ ge t_ \ star，$ for somo $ t_ \ star> 0）$用于非线性schrödinger问题的解决方案$ {在一个有限域$ \ mathbb {r}^n，$ $ n \ le 3，$ $ $ a \ in \ mathbb {c} $带有$ \ im（a）> 0 $（damping案例）和在关键的假设下m \，\ im（a）\ ge（1-m）| \ re（a）|。$我们使用能量方法以及几个先验估计来证明主要结论。方程式中非lipschitz非线性项的存在引入了解决方案的缺乏，需要根据数据所假定的规律性研究某些不同感觉的解决方案的存在和独特性。
We prove the \textit{finite time extinction property} $(u(t)\equiv 0$ on $Ω$ for any $t\ge T_\star,$ for some $T_\star>0)$ for solutions of the nonlinear Schrödinger problem ${\rm i} u_t+Δu+a|u|^{-(1-m)}u=f(t,x),$ on a bounded domain $Ω$ of $\mathbb{R}^N,$ $N\le 3,$ $a\in\mathbb{C}$ with $\Im(a)>0$ (the damping case) and under the crucial assumptions $0<m<1$ and the dominating condition $2\sqrt m\,\Im(a)\ge(1-m)|\Re(a)|.$ 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.
