Boundary-value problem for the Kortewegde Vries-Burgers type equation

We consider the initial{boundary-value problem on the half-line for the Korteweg{ de Vries equation ut + uux + uxxx = 0; t> 0; x> 0; u(x; 0) = u0(x); x> 0; u(0; t) = 0; t> 0: 9>=>; We prove that if the initial data u0 2H0;21 \H0;3=22 and the norm ku0kH 0;21 + ku0kH 0;3=22 are su±ciently small, where Hs;kp = ff 2 L2; kfkH s;kp