Asymptotics for the Korteweg-de Vries-Burgers Equation  

作　　者：Nakao HAYASHI Pavel I.NAUMKIN 

机构地区：[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

出　　处：《Acta Mathematica Sinica,English Series》2006年第5期1441-1456,共16页数学学报（英文版）

基　　金：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

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

分 类 号：O175[理学—数学]