兩分量Novikov方程的爆破準(zhǔn)則和持續(xù)性

陳 涵, 嚴(yán)可欣, 蔣先江

兩分量Novikov方程的爆破準(zhǔn)則和持續(xù)性

陳 涵, 嚴(yán)可欣, 蔣先江*

（寧波大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 浙江 寧波 315211）

兩分量Novikov系統(tǒng); 柯西問(wèn)題; 爆破準(zhǔn)則; 持續(xù)性

本文研究下列兩分量Novikov系統(tǒng):

這個(gè)方程最早是由Novikov[4]在研究非局部偏微分方程的對(duì)稱分類中得出的. Novikov方程具有非常豐富的研究結(jié)果, Hone和Wang等[5-6]證明了其具有雙哈密頓結(jié)構(gòu)和無(wú)窮守恒律, 并且完全可積. 進(jìn)一步, 關(guān)于方程(2)式在Besov空間和Soblev空間的局部適定性、持續(xù)性、解的爆破現(xiàn)象和全局存在性等結(jié)論可以參考文獻(xiàn)[7-10].

另外一個(gè)兩分量Novikov系統(tǒng)稱之為Geng- Xue方程:

1 爆破條件

首先, 回憶方程組(1)式的局部適定性結(jié)論, 通過(guò)Littlewood-Paley定理及Besov空間定義性質(zhì), 得到在Sobolev空間上局部適定性的結(jié)果如下:

下面給出Morse-type估計(jì)和一維運(yùn)輸方程一些常用的引理.

引理1[17]

引理2[17]考慮如下線性運(yùn)輸方程的初值問(wèn)題:

其中,

在文獻(xiàn)[2]中得到已有的爆破結(jié)論:

其中,

借助上面的估計(jì)和引理1有

代入(11)式, 得到

同樣地,

結(jié)合式(14)和(15), 應(yīng)用Gronwall不等式得到

可推斷出

利用引理1以及式(12)和(13), 有下列估計(jì):

代入(18)式, 得到

同樣地,

利用引理1以及式(12)和(13), 有下面的估計(jì):

同樣地,

結(jié)合式(23)和(24), 應(yīng)用Gronwall不等式

或

根據(jù)式(28)和Sobolev嵌入不等式, 有

這和定理3產(chǎn)生矛盾.

另一方面, 利用Sobolev嵌入定理, 若

或

2 持續(xù)性

首先, 給出一些輔助函數(shù)定義[19]:

注如果取一般的標(biāo)準(zhǔn)權(quán)重函數(shù):

證明 將方程組(1)改寫成弱解形式:

對(duì)式(35)每項(xiàng)作估計(jì)

另一方面從定理5的證明中, 容易得到

將上面2個(gè)估計(jì)代入(44)式

同樣地,

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Blow-up criterion and persistence properties for a two-component Novikov system

CHEN Han, YAN Kexin, JIANG Xianjiang*

( School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China )

two-component Novikov system; Cauchy problem; blow-up criterion; persistence properties

O175

A

1001-5132（2023）03-0036-07

2022?07?08.

寧波大學(xué)學(xué)報(bào)（理工版）網(wǎng)址: http://journallg.nbu.edu.cn/

浙江省自然科學(xué)基金（LY22A010005）.

陳涵（1998－）, 女, 浙江寧波人, 在讀碩士研究生, 主要研究方向: 可積系統(tǒng). E-mail: 354186140@qq.com

通信作者:蔣先江（1976－）, 男, 浙江象山人, 講師, 主要研究方向: 偏微分方程. E-mail: jiangxianjiang@nbu.edu.cn

（責(zé)任編輯 章踐立）