Asymptotics of solutions to the boundary-value problem for the Korteweg-de Vries-Burgers equation on a half-line

Nakao Hayashi, Elena I. Kaikina, Ilia A. Shishmarev

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13 Citations (Scopus)

Abstract

We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers equation, ut + ( -1)α uux - uxx + (-1)α uxxx = 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 H0,ω ∩ H1,0, where Hs,k = {f ε L2; ∥f∥Hsk = ∥ 〈x〉k 〈i∂xs f∥L2 < ∞}, ω ε (1/2, 3/2), and the norm ∥u0H0 ω + ∥u0H1,0 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 x1+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.

Original languageEnglish
Pages (from-to)343-370
Number of pages28
JournalJournal of Mathematical Analysis and Applications
Volume265
Issue number2
DOIs
Publication statusPublished - 2002 Jan 15

Keywords

  • Dissipative nonlinear evolution equation
  • Half-line
  • Korteweg-de Vries-Burgers equation
  • Large time asymptotics

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