Non-homogeneous boundary value problems of the Kawahara equation posed on a finite interval

Abstract

Considered in this paper is the initial boundary value problem (IBVP) of the Kawahara equation, a class of the fifth order KdV equation, posed on a finite interval, (0.1)ut+ux+βuxxx+uxxxxx+uux=0,0<x<L,t>0,u(x,0)=ϕ(x), subject to the non-homogeneous boundary conditions (0.2)Bju=hj(t),j=1,2,3,4,5t>0where Bju=∑k=04(ajk∂xku(0,t)+bjk∂xku(L,t)),j=1,2,3,4,5and ajk,bjk(k=0,1,2,3,4andj=1,2,3,4,5) are real constants. Under some general assumptions imposed on the coefficients ajk,bjk, the IBVP (0.1)–(0.2) is shown to be locally well posed in the space Hs(0,L) for any s≥0 with naturally compatible ϕ∈Hs(0,L) and boundary values hj,j=1,2,3,4,5 belonging to some appropriate spaces with optimal regularity. The sharp Kato smoothing properties (due to Kenig, Ponce and Vega (Kenig et al., 1991; Kenig et al., 1993)) of the pure initial value problem (IVP) of the linear inhomogeneous fifth order KdV equation posed on the whole line R, vt+βvxxx+vxxxxx=g(x,t),v(x,0)=ψ(x),x,t∈R,have played an important role in establishing the well-posedness of the IBVP (0.1)–(0.2).