非线性梁方程的适定性与不适定性
Well-posedness and Ill-posedness for the Nonlinear Beam Equation
摘要：
本文研究了具有初始数据$f\in\dot{H}^s, g\in\dot{H}^{s-2}$和$f\in H^s, g\in H^{s-2}$的非线性梁方程的Schmidhartz估计.推广了H. Lindblad和C. D.Sogge [10]和T. Cazenave和F. B. Weissler [4]对非线性梁方程进行了改进，以确定证明适定性和散射结果所需的最小正则性，这些结果具有低正则性数据。最后，我们还利用M. Christ，J. Colliander和T. Tao [2]证明了当$ω=-1$.
英文摘要：
We investigate Strichartz estimates for the nonlinear beam equation with initial data $f\in\dot{H}^s, g\in\dot{H}^{s-2}$ and $f\in H^s, g\in H^{s-2}$. We extend results of H. Lindblad and C. D.Sogge [10] and T. Cazenave and F. B. Weissler [4] to nonlinear beam equations to determine the minimal regularity that is needed to prove well-posedness and scattering results with low regularity data. Finally, we also use small dispersion analysis of M. Christ, J. Colliander and T. Tao [2] to prove the nonlinear beam equation is ill-posed in defocusing case $ω=-1$ when $ 0<s<s_c=\frac{n}{2}-\frac{4}{κ-1}$.
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