Asymptotics for the Korteweg-de Vries-Burgers equation

We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation u_t + uu_x ? u_xx + u _xxx = 0, x ε R, t > 0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u ₀ ε H^s (R) ∩ L ¹ (R), where s > - 1/2, then there exists a unique solution u (t, x) ε C^∞ ((0,∞) ;H ^∞ (R)) to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u(t) = t^-1/2 f_M (( ·)t^-1/2) + o(t ^-1/2) as t → ∞, where f _M is the self-similar solution for the Burgers equation. Moreover if xu ₀ (x) L ¹ (R) , then the asymptotics are true u(t) = t^-1/2 f _M (( ·)t^-1/2) + O(t^-1/2-γ) where γ ε (0,1/2).