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

Abstract

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 space Hs(0,1) for s∈[0,5]; Lastly, we prove that the solution of the closed loop system (0.1) is exponentially decay in the space Hs(0,1) for s∈[0,5].