论文标题
阈值解决方案的限制性不均匀NLS
Threshold solutions for the intercritical inhomogeneous NLS
论文作者
论文摘要
我们考虑$ h^1（\ Mathbb {r}^3）$，\ begin {equation} i \ partial_t u + + + + + | x | x | x |^{ - b} |^{2} | |^{2} u = 0，先前的工作已经在基态解决方案$ q $确定的质量能阈值下建立了爆炸/散射二分法。 在这项工作中，我们正好研究了这个质量能阈值的解决方案。除了基态解决方案外，我们还证明了解决方案$ q^\ pm $的存在，该$ q^\ pm $沿正时方向接近常驻波，但要么在负时间方向上爆炸或散布。使用这些特定的解决方案，我们将所有可能的行为分类用于阈值解决方案。特别是，该解决方案要么像子阈值案例一样行为，要么与$ e^{it} q $，$ q^+$或$ q^ - $一致。
We consider the focusing inhomogeneous nonlinear Schrödinger equation in $H^1(\mathbb{R}^3)$, \begin{equation} i\partial_t u + Δu + |x|^{-b}|u|^{2}u=0,{equation} where $0 < b <\tfrac{1}{2}$. Previous works have established a blowup/scattering dichotomy below a mass-energy threshold determined by the ground state solution $Q$. In this work, we study solutions exactly at this mass-energy threshold. In addition to the ground state solution, we prove the existence of solutions $Q^\pm$, which approach the standing wave in the positive time direction, but either blow up or scatter in the negative time direction. Using these particular solutions, we classify all possible behaviors for threshold solutions. In particular, the solution either behaves as in the sub-threshold case, or it agrees with $e^{it}Q$, $Q^+$, or $Q^-$ up to the symmetries of the equation.
