Asymptotics for the Korteweg-de Vries-Burgers Equation Asymptotics for the Korteweg–de Vries–Burgers Equation期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Asymptotics for the Korteweg-de Vries-Burgers Equation

引用本文：

Nakao HAYASHI Pavel I. NAUMKIN. Asymptotics for the Korteweg-de Vries-Burgers Equation[J]. 数学学报(英文版), 2006, 22(5): 1441-1456. DOI: 10.1007/s10114-005-0677-3

作者姓名：

Nakao HAYASHI Pavel I. NAUMKIN

作者单位：

[1]Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka, 560-0043, Japan [2]Instituto de Matemdticas, UNAM Campus Morelia, AP 61-3 （Xangari）, Morelia CP 58089, Michoacán, Mexico

基金项目：

The work of N. H. is partially supported by Grant-In-Aid for Scientific Research （A）（2）（No. 15204009）, JSPS and The work of P. I. N. is partially supported by C0NACYT.

摘要：

We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut＋uux-uxx＋uxxx=0,x∈R,t〉0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u0 ∈H^s （R）∩L^1 （R）, where s 〉 -1/2, then there exists a unique solution u （t, x） ∈C^∞ （（0,∞）;H^∞ （R）） to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u（t）=t^-1/2fM（（·）t^-1/2）＋0（t^-1/2） as t →∞, where fM is the self-similar solution for the Burgers equation. Moreover if xu0 （x） ∈ L^1 （R）, then the asymptotics are true u（t）=t^-1/2fM（（·）t^-1/2）＋O（t^-1/2-γ） where γ ∈ （0, 1/2）.

关键词：

渐近性初始数据数学分析 Korteweg-de Vries-Burgers方程

收稿时间：

2004-03-04

修稿时间：

2004-03-042005-04-11

Asymptotics for the Korteweg–de Vries–Burgers Equation

Nakao Hayashi,Pavel I. Naumkin. Asymptotics for the Korteweg–de Vries–Burgers Equation[J]. Acta Mathematica Sinica(English Series), 2006, 22(5): 1441-1456. DOI: 10.1007/s10114-005-0677-3

Authors:

Nakao Hayashi Pavel I. Naumkin

Affiliation:

(1) Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka, 560–0043, Japan;(2) Instituto de Matemáticas, UNAM Campus Morelia, AP 61–3 (Xangari), Morelia CP 58089, Michoacán, Mexico

Abstract:

We study large time asymptotics of solutions to the Korteweg–de Vries–Burgers equation

$u_{t} + uu_{x} - u_{{xx}} + u_{{xxx}} = 0,x in {text{R}},t > 0.$

We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u ₀ ∈ H ^s (R) ∩ L ¹ (R) , where $s > - frac{1} {2},$ then there exists a unique solution u (t, x) ∈ C ^∞ ((0,∞) ;H ^∞ (R)) to the Cauchy problem for the Korteweg–de Vries–Burgers equation, which has asymptotics

$u{left( t right)} = t^{{ - frac{1} {2}}} f_{M} {left( {{left( cdot right)}t^{{ - frac{1} {2}}} } right)} + o{left( {t^{{ - frac{1} {2}}} } right)}$

as t → ∞, where f _M is the self–similar solution for the Burgers equation. Moreover if xu ₀ (x) ∈ L ¹ (R) , then the asymptotics are true

$u{left( t right)} = t^{{ - frac{1} {2}}} f_{M} {left( {{left( cdot right)}t^{{ - frac{1} {2}}} } right)} + O{left( {t^{{ - frac{1} {2} - gamma }} } right)},$

where $gamma in {left( {0,frac{1} {2}} right)}.$ The work of N. H. is partially supported by Grant–In–Aid for Scientific Research (A)(2) (No. 15204009), JSPS and The work of P. I. N. is partially supported by CONACYT

Keywords:

Korteweg-de Vries-Burgers equation asymptotics for large time large initial data

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