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

We establish the boundary smoothing properties for the linear Kawahara equation:equation(0.1){∂tu−∂x5u+β∂x3u=0,x∈(0,1),t⩾0,u(x,0)=0,x∈(0,1),u(0,t)=h1(t),u(1,t)=h2(t),∂xu(0,t)=h3(t),∂xu(1,t)=h4(t),∂x2u(1,t)=h5(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=1512πi∫r−i∞r+i∞estΔj(s)Δ(s)eλj(s)xds. Then, by the fine estimates on Δj(s)Δ(s) (j=1,2,3,4,5j=1,2,3,4,5), we establish the boundary smoothing effect: for any s⩾0s⩾0, if the boundary data (h1(t),h2(t),h3(t),h4(t),h5(t))∈H0s+25(R+)×H0s+25(R+)×H0s+15(R+)×H0s+15(R+)×H0s5(R+), then the solution u∈C(R+;Hs(0,1))∩L2(R+;H2+s(0,1))u∈C(R+;Hs(0,1))∩L2(R+;H2+s(0,1)) and possesses the sharp trace regularity∂xku∈C([0,1];Hs+2−k5(R+))fork=0,1,2.