We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut + uux ? uxx + 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 Hs (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 fM (( ·)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).
- Asymptotics for large time
- Korteweg-de Vries-Burgers equation
- Large initial data
ASJC Scopus subject areas
- Applied Mathematics