Global well-posedness and exponential decay for fifth-order Korteweg–de Vries equation posed on the finite domain

摘要

This paper studies the fifth-order Korteweg–de Veries equation posed on finite interval(0.1){?tu??x5u+ε1?x4u+κ1?x3u?ε2?x2u+κ2?xu+ε3u+u?xu=0,x∈(0,1),t≥0,u(x,0)=Φ(x),x∈(0,1),u(0,t)=u(1,t)=?xu(0,t)=?xu(1,t)=0,?x2u(1,t)=ξ?x2u(0,t),t≥0.Firstly, we establish smoothing effect of Kato type for the fifth-order Korteweg–de Veries equation posed on finite interval; Secondly, we show that the closed loop system(0.1)is globally well-posed in the spaceHs(0,1)fors∈[0,5]; Lastly, we prove that the solution of the closed loop system(0.1)is exponentially decay in the spaceHs(0,1)fors∈[0,5].