论文标题
具有sublrinear非线性的严重阻尼schr {Ö} dinger方程的有限时间灭绝
Finite time extinction for a critically damped Schr{ö}dinger equation with a sublinear nonlinearity
论文作者
论文摘要
本文完成了一些作者先前的一些研究，当时非线性schr {Ö} dinger方程的有限时间灭绝，当时非线性阻尼术语对应于某些``饱和的非kerr law''$ f（| u |^2）u |^2）U = = \ frac {a} a} a} a} { $ a \ in \ mathbb {c}，$ $ \ varepsilon \ geqslant0，$ $2α=（1-m）$和$ m \ in [0,1）。$在这里我们考虑sublinear案例$ 0 <m <m <1 $，带有一个重要的湿damped cofficity：$ a \ in \ mathbb { $ d（m）= \ big \ {z \ in \ mathbb {c}; \; \ Mathrm {im}（z）> 0 \ text {and} 2 \ sqrt {m} \ mathrm {im}（im}（z）=（1-m）\ Mathrm {re}（z）\ big \}。 Liskevich和Perel'muter [16]以及Cialdea和Maz'ya [14]的最新研究）。在获得适当的先验估计值后，通过合适的能量法证明了解决方案的有限时间灭绝。大多数结果适用于非必要的空间域。
This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schr{ö}dinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{\varepsilon+(|u|^2)^α}u,$ with $a\in\mathbb{C},$ $\varepsilon\geqslant0,$ $2α=(1-m)$ and $m\in[0,1).$ Here we consider the sublinear case $0<m<1$ with a critical damped coefficient: $a\in\mathbb{C}$ is assumed to be in the set $D(m)=\big\{z\in\mathbb{C}; \; \mathrm{Im}(z)>0 \text{ and } 2\sqrt{m}\mathrm{Im}(z)=(1-m)\mathrm{Re}(z)\big\}.$ Among other things, we know that this damping coefficient is critical, for instance, in order to obtain the monotonicity of the associated operator (see the paper by Liskevich and Perel'muter [16] and the more recent study by Cialdea and Maz'ya [14]). The finite time extinction of solutions is proved by a suitable energy method after obtaining appropiate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.
