Asymptotics of Solutions to the Boundary-Value Problem for the Korteweg–de Vries–Burgers Equation on a Half-Line

We study the following initial–boundary value problem for the Korteweg–de Vries–Burgers equation, 0 1 ut + −1 αuux − uxx + −1 αuxxx = 0 x t ∈ R+ × R+ u x 0 = u0 x x∈ R+ ∂n xu 0 t = 0 n= 0 α t ∈ R+ where α = 0 1. We prove that if the initial data u0 ∈ H0 ω ∩ H1 0, where Hs k = f ∈ L2 f Hs k = x k i∂x sf L2 < ∞ , ω ∈ 1 2 3 2 , and the norm u0 H0 ω + u0 H1 0 is sufficiently small, then there exists a unique solution u ∈ C 0 ∞ H0 ∩ C 0 ∞ H1 ω of the initial–boundary value problem (0.1), 343 0022-247X/02 $35.00  2002 Elsevier Science All rights reserved. 344 hayashi, kaikina, and shishmarev where ∈ 0 1 2 . Moreover, if the initial data are such that x1+μu0 x ∈ L1, μ = ω − 1 2 , then there exists a constant A such that the solution has the asymptotics u x t = A t α x 2√t t + O min x √t 1 t−1− μ2 for t →∞uniformly with respect to x > 0 where α = 0 1, 0 q t = q/√π e−q2 , 1 q t = 1/2√π√t e−q2 2q√t − 1 + e−2q√t .