修正Kawahara方程的收敛问题与色散爆破

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 595-608.

修正Kawahara方程的收敛问题与色散爆破

王伟敏(),闫威^*()

河南师范大学数学与信息科学学院河南新乡 430007

收稿日期:2023-08-16 修回日期:2024-01-02 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *闫威, Email: 011133@htu.edu.cn
作者简介:王伟敏, Email: ydn1129@163.com
基金资助:
河南省骨干教师项目(2017GGJS044)

Convergence Problem and Dispersive Blow-up for the Modified Kawahara Equation

Wang Weimin(),Yan Wei^*()

School of Mathematics and Information Science, Henan Normal University, Henan Xinxiang 453007

Received:2023-08-16 Revised:2024-01-02 Online:2024-06-26 Published:2024-05-17
Supported by:
Young Core Teachers Problem of Henan Province(2017GGJS044)

摘要/Abstract

摘要：

该文主要研究修正 Kawahara 方程的收敛问题与色散爆破. 首先, 利用傅里叶限制范数法, 高低频分解技巧以及 Strichartz 估计, 用三种不同的方法证明在空间 $ H^{s}(\mathbb{R}) $ $ (s\geq\frac{1}{4}) $ 中, 对几乎处处的 $ x\in\mathbb{R} $, 当 $ t\rightarrow 0 $ 时, $ u(x,t)\rightarrow u_0(x) $, 其中 $ u(x,t) $ 是修正 Kawahara 方程的解, $ u_0(x) $ 是其柯西问题的初值. 其次, 利用三线性估计和傅里叶限制范数法, 证明在空间 $ H^{s}(\mathbb{R}) $ $ (s>0) $ 中, 当 $ t\rightarrow 0 $ 时, $ u(x,t)\rightarrow U(t)u_0(x) $ (与 $ x $ 无关). 最后, 给出方程解的色散爆破.

关键词: 修正 Kawahara 方程, 逐点收敛, 一致收敛, 色散爆破

Abstract:

In this paper, we consider the convergence problem and dispersive blow-up for the modified Kawahara equation. Firstly, we prove that $ u(x,t)\rightarrow u_0(x),$ a.e. $ x\in\mathbb{R} $ as $ t\rightarrow 0 $ by the Fourier restriction norm method, high-low frequency technique and Strichartz estimate, respectively. Here $ u(x,t) $ is the solution of the modified Kawahara equation, and the initial value $ u_0(x)\in H^{s}(\mathbb{R}) $ $ (s\geq\frac{1}{4}) $. Secondly, using the Fourier restriction norm method, we show that $ u(x,t)\rightarrow U(t)u_0(x) $ as $ t\rightarrow 0 $ with $ u_0(x)\in H^{s}(\mathbb{R}) $ $ (s>0) $. Finally, we establish the dispersive blow-up of the modified Kawahara equation.

Key words: Modified Kawahara equation, Pointwise convergence, Uniform convergence, Dispersive blow-up

中图分类号:

O175.2

王伟敏, 闫威. 修正Kawahara方程的收敛问题与色散爆破[J]. 数学物理学报, 2024, 44(3): 595-608.

Wang Weimin, Yan Wei. Convergence Problem and Dispersive Blow-up for the Modified Kawahara Equation[J]. Acta mathematica scientia,Series A, 2024, 44(3): 595-608.

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