广义bbmburgers方程初边值问题解的渐近行为.docx

性波也(。)和好(。,t)($,t)=咖2(z)+镏(岳,t)
本文共分为两章: 第一章,,在回
顾前人工作的基础上,叙述了本文的结果,并指出方法的不同之处. 第二章,.
第一节讨论了情形(1)和(2),第二节讨论了情形(4)和(5),第三节讨论了情形
(3),均得到了解的整体存在性及渐近行为.
关键词:广义BBM—Burgers方程,L2能量方法,先验估计,驻波解,稀疏波, 渐近行为.
硕士学位论文
\1、、1 FR’S¨jESlN
ABSTRACT
In this paper,we consider the initial-boundary problem for the generalized Benjamin-
Bona-Mahony-Burgers equation(for brevity,we call it BBM-Burgers equation later),
+ ‰z n , 引矿 o >
=
毗吣“,幻‰._亘砒.Ⅻ= 饥 0
, ,
(I)
;』 。
吣b 幻 b卸= 眦茹
Where/(n)is a smooth convex function defined on R and u士are two given constants. Under the assumption ofu一<“+,we study the global existence and asymptotic behavior of the solution as t.+oo for(I)..,For the initial-boundary problem(I),as in【10], the signs of the characteristic speeds,’(u±)divide the asymptotic state into five cases:
(1),’(t‘一)<,7(“+)<0
(2),’(¨一)<,7(“+)=0,
(3),’(“一)<0<ft(“+),
(4)0=,’(¨一)<fl(“+),
(5)0<,’(u一)<f r∞+).
We also get the global existence with some suitable restriction to the initial .
over,for the case(1)and(2),sup It‘(z,t)一也(z)I-÷0 as t_÷oo,where也(。)a=1,2)
xER+
is the stationary solution of(I),which satisfy:
f,(咖)。=札。,。∈R+,
{
【曲(o)=“一,咖(+oo):=t‘+.
in
=u冗(茁/t)lR+,and uR(茁/亡)lR+is the rarefaction wave of the Riemann problem
f饥+,(u)z=o,
卜删㈤=㈦≥
For the case(3),sup 1t‘(z,t)一西3(茁,t)j叶0,as t_÷。。,where圣3(z,t)is the
superpbsition of咖2(z)and啪拿(茁,t),and垂3(z,t)=≯2(z)+妒擎(皿,t).
This paper is made up of two chapters.
In chapter one,we introduce the background of generalized BBM—Burgers equation and the relevant research progress,furthermore,we state our main results following from some retrospection of the results obtained by previous mathematicians.
The chapter two is divided into three sections,In section 1,we study the cases(1)
and(2).In section 2,we study the cases(4)and(5).In section 3,we study the case(3). For the five Eases,we all get the global existence and asymptotic behavior