有限区间上五阶 KdV 方程初始边界值问题的全局好求解性

IF 1 4区 数学 Q1 MATHEMATICS Open Mathematics Pub Date : 2023-12-16 DOI:10.1515/math-2023-0158
Xiangqing Zhao, Chengqiang Wang, Jifeng Bao
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Math. Anal. Appl. 470 (2019), 251–278]. A question arises naturally: <jats:italic>Can the local solution be extended to a global one?</jats:italic> This article will address this question. 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引用次数: 0

摘要

我们已经确定了 (0) 的局部解的存在性和唯一性。1) ∂ t u + ∂ x 5 u - u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x <;1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t >;0 ,left\{begin{array}{ll}{\partial }_{t}u+{\partial }_{x}^{5}u-u{\partial }_{x}u=0,& 0\lt x\lt 1,\hspace{1.0lt x\lt 1,u\left(0,t)={h}_{1}\left(t),u\left(1,t)={h}_{2}\left(t),\hspace{0.33em} {\partial }_{x}u\left(1,t)={h}_{3}\left(t),&\ {\partial }_{x}u\left(0,t)={h}_{4}\left(t),\hspace{0.33em}{partial }_{x}^{2}u\left(1,t)={h}_{5}\left(t),& t\gt 0,\end{array}\right. in the study of Zhao and Zhang [Non-homogeneous boundary value problem of the fifth-order KdV equations posed on a bounded interval, J. Math.Anal.Appl. 470 (2019),251-278]。一个问题自然而然地产生了:局部解能否扩展为全局解?本文将探讨这个问题。首先,通过一系列逻辑推导,建立一个全局先验估计,然后将局部解自然扩展为全局解。
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Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
We have established the existence and uniqueness of the local solution for (0.1) t u + x 5 u u x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , x u ( 1 , t ) = h 3 ( t ) , x u ( 0 , t ) = h 4 ( t ) , x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 , \left\{\begin{array}{ll}{\partial }_{t}u+{\partial }_{x}^{5}u-u{\partial }_{x}u=0,& 0\lt x\lt 1,\hspace{1.0em}t\gt 0,\\ u\left(x,0)=\varphi \left(x),& 0\lt x\lt 1,\\ u\left(0,t)={h}_{1}\left(t),u\left(1,t)={h}_{2}\left(t),\hspace{0.33em}{\partial }_{x}u\left(1,t)={h}_{3}\left(t),& \\ {\partial }_{x}u\left(0,t)={h}_{4}\left(t),\hspace{0.33em}{\partial }_{x}^{2}u\left(1,t)={h}_{5}\left(t),& t\gt 0,\end{array}\right. in the study of Zhao and Zhang [Non-homogeneous boundary value problem of the fifth-order KdV equations posed on a bounded interval, J. Math. Anal. Appl. 470 (2019), 251–278]. A question arises naturally: Can the local solution be extended to a global one? This article will address this question. First, through a series of logical deductions, a global a priori estimate is established, and then the local solution is naturally extended to a global solution.
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Open Mathematics
Open Mathematics MATHEMATICS-
CiteScore
2.40
自引率
5.90%
发文量
67
审稿时长
16 weeks
期刊介绍: Open Mathematics - formerly Central European Journal of Mathematics Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant, original and relevant works in all areas of mathematics. The journal provides the readers with free, instant, and permanent access to all content worldwide; and the authors with extensive promotion of published articles, long-time preservation, language-correction services, no space constraints and immediate publication. Open Mathematics is listed in Thomson Reuters - Current Contents/Physical, Chemical and Earth Sciences. Our standard policy requires each paper to be reviewed by at least two Referees and the peer-review process is single-blind. Aims and Scope The journal aims at presenting high-impact and relevant research on topics across the full span of mathematics. Coverage includes:
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