Non-homogeneous boundary value problems of the Kawahara equation posed on a finite interval. (February 2023) - Sriskandasingam, Mayuran; Sun, Shu-Ming; Zhang, Bing-Yu
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) 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).