Asymptotics for the Korteweg-de Vries-Burgers Equation

We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equationut+uux-uxx+uxx=0,x ∈ R,t>0.We are interested in the large time asymptotics for the case when the initial data have an arbitrarysize. We prove that if the initial data u0∈ Hs(R)∩ L1(R), where s >-1/2, then there exists a uniquesolution u(t,x)∈ C∞((0,∞);H∞(R))to the Cauchy problem for the Korteweg-de Vries-Burgersequation, which has asymptoticsu(t)=t-1/2fM((·)t-1/2+o(t-1/2)as t→∞,where fM is the self-similar solution for the Burgers equation. Moreover if xu0(x)∈L1(R),then the asymptotics are trueu(t)=t-1/2fM((·)t-1/2+O(t-1/2-γ),where γ∈(0,1/2).