Non homogeneous Dirichlet problem for the KdVB equation on a segment

We study the Non homogeneous Dirichlet problem with large initial data for the KdVB equation on the interval x ∈ (0,1) ⎪⎪⎨ ⎪⎪⎩ ut +uxu−uxx +uxxx = 0, t > 0, x ∈ (0,1) u(x,0) = u0(x), x ∈ (0,1) u(0,t) = u(1,t) = 0, t > 0 ux(1,t) = h(t), t > 0. (1) We prove that if the initial data u0 ∈ L2 and boundary data h(t) ∈ H∞(0,∞) then there exist a unique solution u ∈ C([0,∞) ;L2)∪C((0,∞) ;H1) of the initial-boundary value problem (1). We also obtain the large time asymptotic of solution uniformly with respect to x ∈ (0,1) as t → ∞. Mathematics subject classification (2010): 35Q35.