On the boundary-value problem for the Korteweg-de Vries equation

Nakao Hayashi, Elena I. Kaikina, J. L.Guardado Zavala

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11 Citations (Scopus)

Abstract

We consider the initial-boundary-value problem on the half-line for the Kortewegde Vries equation ut + uux + uxxx = 0, t > 0, x > 0,} u(x, 0) = u0(x), x > 0, u(0, t) = 0, t > 0. We prove that if the initial data u0 ∈ H1 0,2 ∩ H20,3/2 and the norm ∥u 0H10,2 + ∥u0H20,3/2 are sufficiently small, where Hps,k = {f ∈ L 2; ∥f∥Hps,k = ∥〈x〉 k〈i∂xsf∥Lp < ∞}, 〈x〉 = √1 + x2, then there exists a unique solution u ∈ C([0, ∞), H20,1) ∩ L (0, ∞, H20,3/2) ∩ C((0, ∞), H22,0) ∩ L(0, ∞, H23,0) of the initial-boundary-value problem. Moreover, we proved that there exists a constant B such that the solution has the following asymptotics, u(x, t) = t-1Bx/3√tAi(x/3√t) + O(t-1-δ/3 max(1,x/3√t)) for t → ∞ uniformly with respect to x > 0, where 0 < δ < 1/2, Ai(q) is the Airy function defined by Ai(q) = ∫-i∞i∞ e -z3+zq dz.

Original languageEnglish
Pages (from-to)2861-2884
Number of pages24
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume459
Issue number2039
DOIs
Publication statusPublished - 2003 Nov 8
Externally publishedYes

Keywords

  • Half-line
  • Korteweg-de Vries equation
  • Large-time asymptotics
  • Nonlinear evolution equation

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

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