THE INITIAL-VALUE PROBLEM FOR THE FORCED BBM EQUATION

• Main Authors: Chia-Jung Kuo, 郭佳蓉
• Other Authors: Tai-Cheng Tso
• Format: Others
• Language: en_US
• Published: 1999
• Online Access: http://ndltd.ncl.edu.tw/handle/94299389333390489771

碩士 === 國立高雄師範大學 === 數學系 === 87 === This paper is concerned with the assocoiated linear IVP $$ u_t+(vu)_x-u_{xxt}= f(x,t),$$ $$ u(x,0)= u_0(x).$$ for $x\in {\bf R}$, $t>0,$ where $v=v(x,t)$ is a given function. The well-posedness of the IVP in the space ${\cal H}_T\cap{\cal C}_T^{2,1}$ (for a given finite $T>0$) is established and estimates of the solution in terms of $u_0$ and $f$ are provided. We use the method of linear estimates to establish the analyticity and well-posedness of the forced Benjamin-Bona-Mahony (BBM for short) equation $$ u_t+uu_x-u_{xxt} = f(x,t),$$ $$ u(x,0)=u_0(x). $$ for $x\in{\bf R},\;t>0$ under the conditions on the initial value $u_0(x) \in H^1({\bf R}) \cap {\cal C}_b^2({\bf R})$ and the forcing function $f(x,t) \in {\cal C}_T \cap {\cal L}_T$ are provided.