A Class Of Kdv-burgers Equation As A Whole Posedness

We consider the Cauchy problem for the generalized Korteweg-de Vries-Burgers equation with initial data,where, is the Fourier multiplier associated with the symbol |￡|^2α. For the generalized KdV-Burgers equation, we study the well-posed and ill-posed problem with initial data in the homogeneous Sobolev Space H^s and Sobolev Space H^s, separatively.To solove the well-posedness of the equation mentioned above, the main difficulty is how to get the bilinear estimate. Here , in the Sobolev Space H^s, with s > —α, we get the bilinear estimate using dual method by dispersive relation satisfying a algebraic smoothing relation:When s > —α, in the Sobolev Space H^s , the solution of the generalized KdV-Burgers equation can be proofed to be well-posed by the bilinear estimate and the Fixed Point Theorem.In the homogeneous Sobolev Space H^s, we get the bilinear estimate by dyadic decomposition and abstract multiplier principle by T.Tao. When homogeneous Sobolev Space H^s, the solution of the generalized KdV-Burgers equation can be proofed to be well-posed also by the bilinear estimate and the Fixed Point Theorem. Correspondingly, when , the solution of this equation can be proofed to be ill-posed by dimensional analysis.