论文标题
立方四阶非线性schrödinger方程的随机数据理论
Random data theory for the cubic fourth-order nonlinear Schrödinger equation
论文作者
论文摘要
我们考虑立方非线性四阶方程\ [i \ partial_t u-Δ^2 u +μΔU= \ pm | u |^2 U，\ \quadμ\ geq 0 \]在$ \ mathbb {r}^n，n \ geq 5 $具有随机初始数据。我们证明，在缩放临界规律上几乎可以肯定的是本地良好性。我们还证明了概率的小数据全球适应性和散射。最后，我们证明了全球良好的和散射，具有很大的概率，可以在扩张的立方体上随机进行初始数据。
We consider the cubic nonlinear fourth-order Schrödinger equation \[ i\partial_t u - Δ^2 u + μΔu = \pm |u|^2 u, \quad μ\geq 0 \] on $\mathbb{R}^N, N \geq 5$ with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.
