Abstract

We consider the Zakharov-Kutznesov (ZK) equation posed in ${\mathbb{R}}^d,$ with d = 2 and 3. Both equations are globally well-posed in $L^2({\mathbb{R}}^d).$ In this article, we prove local energy decay of global solutions: if u(t) is a solution to ZK with data in $L^2({\mathbb{R}}^d),$ then $$\underset{t\to \infty }{\mathrm{lim\hspace{0.17em}}\inf }{\int }_{{\Omega }_d(t)}{u}^2(\mathbf{x},t)\mathrm{d}\mathbf{x}=0,$$for suitable regions of space ${\Omega }_d(t)\subseteq {\mathbb{R}}^d$ around the origin, growing unbounded in time, not containing the soliton region. We also prove local decay for $H^1({\mathbb{R}}^d)$ solutions. As a byproduct, our results extend decay properties for KdV and quartic KdV equations proved by Gustavo Ponce and the second author. Sequential rates of decay and other strong decay results are also provided as well.

摘要

我们考虑提出的 Zakharov-Kutznesov (ZK) 方程 ${\mathbb{电阻}}^d,$与d  = 2和3这两个方程是在全球范围内适定${升}^2({\mathbb{电阻}}^d).$在本文中，我们证明了全局解的局部能量衰减：如果u ( t ) 是 ZK 的解，其中数据为${升}^2({\mathbb{电阻}}^d),$ 然后 $$\underset{吨\to \infty }{\mathrm{林\hspace{0.17em}}\mathrm{信息}}{\int }_{{\Omega }_d(吨)}{你}^2(\mathbf{X},吨)\mathrm{d}\mathbf{X}=0,$$对于合适的空间区域 ${\Omega }_d(吨)\subseteq {\mathbb{电阻}}^d$围绕原点，随时间无限增长，不包含孤子区域。我们还证明了局部衰减$H^1({\mathbb{电阻}}^d)$解决方案。作为副产品，我们的结果扩展了由 Gustavo Ponce 和第二作者证明的 KdV 和四次 KdV 方程的衰减特性。还提供了连续衰减率和其他强衰减结果。