三次六阶Boussinesq方程的初值和初边值问题

中文摘要 AI 辅助

本文研究具有三次非线性的六阶Boussinesq方程，\\( u_{tt}-u_{xx}+k u_{xxxx}-u_{xxxxxx}+\\( u^3 \\) _{xx}=0, \\ \\ \mbox{其中}k=\pm 1,\\) 在\\(\mathbb{R}^+\\)和\\(\mathbb{R}\\)上提出。在相关的Bourgain型空间\\(X^{s,b}\\)中，对于\\(s>-\frac12\\)且\\(b=\frac12\\)和\\(b>\frac12\\)，局部适定性得到了改进（参见\\(\cite{esfahani2025well,zhong2024local}\\)）。关键要素是通过借鉴Tao的乘子方法、插值理论以及我们早期关于色散方程初边值问题的工作\\(\cite{li2020low,tao2001multilinear}\\)中的思想，细化Bourgain型空间上的三线性估计。

英文摘要

In this paper, we consider the sixth-order Boussinesq equation with a cubic nonlinearity, \[ u_{tt}-u_{xx}+k u_{xxxx}-u_{xxxxxx}+\left( u^3 \right) _{xx}=0, \ \ \mbox{with }k=\pm 1,\] posed on both $\mathbb{R}^+$ and $\mathbb{R}$. The local wellposedness are improved in related Bourgain-type spaces, $X^{s,b}$, for $s>-\frac12$ with $b=\frac12$ and $b>\frac12$ accordingly (See. \cite{esfahani2025well,zhong2024local}). The key ingredient is to refine the trilinear estimate on Burgain type space by adapting ideas from Tao's multiplier method, interpolation theory and our early work on IBVPs of dispersive equations \cite{li2020low,tao2001multilinear}.

发表机构

• University of Electronic Science and Technology of China（电子科技大学）

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