高阶波动方程的时空估计与低能量散射

本文研究了高阶波动方程的低能量散射理论，基本工具是高阶线性波动方程解的时空估计．与经典的二阶波动方程解的时空估计证明不同，我们采用泛函分析的方法与待定指标技巧，首次给出了高阶线性波动方程的时空估计，藉此与非线性函数在齐次Ｓｏｂｏｌｅｖ空间中的估计，获得了高阶波动方程的低能量散射结论．与此同时，也得到了具临界增长的高阶波动方程的柯西问题在低能量条件下的整体存在唯一性．

The Time-Space Estimates and Scattering at Low Energy for Nonlinear Higher order Wave Equations

In this paper we study the scattering theory at low energy for nonlinear higher order wave equations.Our main tools are time-space estimates of linear higher-order wave equation.For this reason we first obtain the time-space estimates for linear higher-order wave equation by the technology of undertermined index and the method of functional Analysis which are different from the method of proof for classical wave equation,further we prove the estimates of nonlinear term in homogenous Sobolev space.Using time-space estimates and nonlinear estimates we obtain the scattering theory at low energy for nonlinear higher-order wave equation.Meanwhiles,we also obtain the global existence and uniqueness for nonlinear higher-order wave equations under condition of low energy.