LAUSR

Abstract This article is concerned with an initial–boundary-value problem (IBVP) for the sixth order Boussinesq equation on a bounded domain with non-homogeneous boundary conditions, (1) u t t − u x x + β u x x x x − u x x x x x x + ( u 2 ) x x = 0 , x ∈ ( 0 , 1 ) , t > 0 , u ( x , 0 ) = φ ( x ) , u t ( x , 0 ) = ψ ( x ) , u x ( 0 , t ) = h 1 ( t ) , u x x x ( 0 , t ) = h 2 ( t ) , u x x x x x ( 0 , t ) = h 3 ( t ) , u x ( 1 , t ) = h 4 ( t ) , u x x x ( 1 , t ) = h 5 ( t ) , u x x x x x ( 1 , t ) = h 6 ( t ) , where β = ± 1 . For 0 ≤ s ≤ 6 , it is shown that the IBVP is locally well-posed if the initial data lie in the product of L 2 -based Sobolev spaces, H s ( 0 , 1 ) × H s − 3 ( 0 , 1 ) , provided the boundary data ( h 1 , h 2 , h 3 ) and ( h 4 , h 5 , h 6 ) lie in the product space, H l o c ( s + 2 ) ∕ 3 ( R + ) × H l o c s ∕ 3 ( R + ) × H l o c ( s − 2 ) ∕ 3 ( R + ) .