论文标题
具有指数非线性的聚焦非线性schrödinger方程的长时间动力学
Long time dynamics for the focusing nonlinear Schrödinger equation with exponential nonlinearities
论文作者
论文摘要
在本文中，我们研究了具有指数非线性的聚焦非线性schrödinger方程 \ [i \ partial_t u +Δu= - \左（e^{4π| u |^2} - 1-4πμ| u |^2 \ right） \ {0，1 \} $。通过使用变分参数，我们首先得出全局存在和有限时间爆炸的不变集。特别是，我们获得了全球存在和有限时间爆炸的尖锐阈值。在$μ= 1 $的情况下，通过调整Arora-Dodson-Murphy \ cite {ADM}的最新论点，我们研究了全球解决方案的长时间动力学。事实证明，要么存在$ t_n \ rightArrow +\ infty $和$ r_n \ righatRow \ infty $，因此$ u（t_n）$在$ b（0，r_n）$中消失了所有$ n \ geq 1 $，或$ h^1 $ in $ n \ geq c。
In this paper, we study the focusing nonlinear Schrödinger equation with exponential nonlinearities \[ i \partial_t u + Δu = - \left(e^{4π|u|^2} - 1 - 4πμ|u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2, \] where $μ\in \{0, 1\}$. By using variational arguments, we first derive invariant sets where the global existence and finite time blow-up occur. In particular, we obtain sharp thresholds for global existence and finite time blow-up. In the case $μ=1$, by adapting a recent argument of Arora-Dodson-Murphy \cite{ADM}, we study the long time dynamics of global solutions. It turns out that either there exist $t_n\rightarrow +\infty$ and $R_n \rightarrow \infty$ such that $u(t_n)$ vanishes inside $B(0,R_n)$ for all $n\geq 1$ or the solution scatters in $H^1$.
