The Cauchy Problem for the Generalized Korteweg-de Vries-Burgers Equation in _H

Abstract

" The Cauchy problem for the generalized Korteweg-de Vries-Burgers equation is considered and the local existence and uniqueness of solutions in L^q(0, T;L^p) \u2229 L^\u221e(0, T; \\dot{H}^{-s})(0 \u2264 s < 1) are obtained for initial data in \\dot{H}^{-s}. Moreover, the local solutions are global if the initial data are sufficiently small in critical case. Particularly, for s = 0, the generalized Korteweg-de Vries-Burgers equation satisfies the energy equality, so the initial data can be arbitrarily large to obtain the global solution."