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

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) u t + u x + β u x x x + u x x x x x + u u x = 0 , 0 < x < L , t > 0 , u (x , 0) = ϕ (x) , subject to the non-homogeneous boundary conditions (0.2) B j u = h j (t) , j = 1 , 2 , 3 , 4 , 5 t > 0 where B j u = ∑ k = 0 4 (a j k ∂ x k u (0 , t) + b j k ∂ x k u (L , t)) , j = 1 , 2 , 3 , 4 , 5 and a j k , b j k (k = 0 , 1 , 2 , 3 , 4 and j = 1 , 2 , 3 , 4 , 5) are real constants. Under some general assumptions imposed on the coefficients a j k , b j k , the IBVP (0.1)–(0.2) is shown to be locally well posed in the space H s (0 , L) for any s ≥ 0 with naturally compatible ϕ ∈ H s (0 , L) and boundary values h j , 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 , v t + β v x x x + v x x x x x = 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). [ABSTRACT FROM AUTHOR]