Boundary smoothing properties of the Kawahara equation posed on the finite domain

We establish the boundary smoothing properties for the linear Kawahara equation:(0.1) { ∂ t u − ∂ x 5 u + β ∂ x 3 u = 0 , x ∈ ( 0 , 1 ) , t ⩾ 0 , u ( x , 0 ) = 0 , x ∈ ( 0 , 1 ) , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 0 , t ) = h 3 ( t ) , ∂ x u ( 1 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t ⩾ 0 in this paper. Firstly, by Laplacian transformation, we give the explicit formula of the solution of (0.1): u ( x , t ) = ∑ j = 1 5 1 2 π i ∫ r − i ∞ r + i ∞ e s t Δ j ( s ) Δ ( s ) e λ j ( s ) x d s . Then, by the fine estimates on Δ j ( s ) Δ ( s ) ( j = 1 , 2 , 3 , 4 , 5 ), we establish the boundary smoothing effect: for any s ⩾ 0 , if the boundary data ( h 1 ( t ) , h 2 ( t ) , h 3 ( t ) , h 4 ( t ) , h 5 ( t ) ) ∈ H 0 s + 2 5 ( R + ) × H 0 s + 2 5 ( R + ) × H 0 s + 1 5 ( R + ) × H 0 s + 1 5 ( R + ) × H 0 s 5 ( R + ) , then the solution u ∈ C ( R + ; H s ( 0 , 1 ) ) ∩ L 2 ( R + ; H 2 + s ( 0 , 1 ) ) and possesses the sharp trace regularity ∂ x k u ∈ C ( [ 0 , 1 ] ; H s + 2 − k 5 ( R + ) ) for k = 0 , 1 , 2 .