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, u_t + ( -1)^α uu_x - u_xx + (-1)^α u_xxx = 0, (x, t) ε R⁺ × R⁺, (0.1) u(x, 0) = u₀(x), x ε R⁺, ∂_xⁿ(0, t) = 0, n = 0, α, t ε R⁺, where α = 0, 1. We prove that if the initial data u₀ ε H^0,ω ∩ H^1,0, where H^s,k = {f ε L²; ∥f∥_H^sk = ∥ 〈x〉^k 〈i∂_x〉^s f∥_L² < ∞}, ω ε (1/2, 3/2), and the norm ∥u₀∥_H⁰ ω + ∥u₀∥_H¹,⁰ is sufficiently small, then there exists a unique solution u ε C([0, ∞), H⁰,^x) ∩ C((0, ∞), H¹,ω) of the initial-boundary value problem (0.1), where x ε (0,1/2). Moreover, if the initial data are such that x^1+μ u₀(x) ε L¹, μ = ω - 1/2, then there exists a constant A such that the solution has the asymptotics for t → ∞ uniformly with respect to x > 0, where α = 0, 1, Φ₀ (q, t) = (q/π)e^-q2, Φ₁ (q,t) 1/2 π t) (e^-q2) (2q t-1) + e^{-2q t}.