论文标题
聚焦四阶非线性schrödinger方程的径向溶液的动力学
Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations
论文作者
论文摘要
我们考虑以下类别的焦点$ l^2 $ - cultical危害四阶非线性schrödinger方程 \ [ i \ partial_t u-δ^2 u +μΔU= - | u |^αu，\ quad（t，x）\ in \ mathbb {r} \ times \ times \ times \ mathbb {r}^n， \] where $N\geq 2$, $μ\geq 0$, and $\frac{8}{N}<α<α^*$ with $α^*:=\frac{8}{N-4}$ if $N\geq 5$ and $α^*=\infty$ if $N\leq 4$.通过使用局部的Morawetz估计值和径向Sobolev嵌入，我们将能量散射在该方程的基态阈值以下，并具有径向对称的初始数据。我们还解决了方程式有限的时间爆破径向溶液的存在。特别是，我们显示出具有径向数据的方程式散射和炸毁的尖锐阈值。我们的散射结果不仅扩大了gu \ cite {guo}证明的结果，其中散射以$μ= 0 $证明，而且还提供了一个替代的简单证明，完全避免使用浓度/紧凑性和刚性参数。在$μ> 0 $的情况下，我们的爆炸结果扩展了Boulenger-lenzmann \ cite {bl}证明的较早结果，其中显示了有限的时间爆炸以用于负能量的初始数据。
We consider the following class of focusing $L^2$-supercritical fourth-order nonlinear Schrödinger equations \[ i\partial_t u - Δ^2 u + μΔu = - |u|^αu, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N, \] where $N\geq 2$, $μ\geq 0$, and $\frac{8}{N}<α<α^*$ with $α^*:=\frac{8}{N-4}$ if $N\geq 5$ and $α^*=\infty$ if $N\leq 4$. By using the localized Morawetz estimates and radial Sobolev embedding, we establish the energy scattering below the ground state threshold for the equation with radially symmetric initial data. We also address the existence of finite time blow-up radial solutions to the equation. In particular, we show a sharp threshold for scattering and blow-up for the equation with radial data. Our scattering result not only extends the one proved by Guo \cite{Guo}, where the scattering was proven for $μ= 0$, but also provides an alternative simple proof that completely avoids the use of the concentration/compactness and rigidity argument. In the case $μ> 0$, our blow-up result extends an earlier result proved by Boulenger-Lenzmann \cite{BL}, where the finite time blow-up was shown for initial data with negative energy.
