ORIGINAL ARTICLES

Acta Mathematica Sinica

, Volume 22, Issue 5, pp 1441-1456

First online:

Asymptotics for the Korteweg–de Vries–Burgers Equation

  • Nakao HayashiAffiliated withDepartment of Mathematics, Graduate School of Science, Osaka University Email author 
  • , Pavel I. NaumkinAffiliated withInstituto de Matemáticas, UNAM Campus Morelia, AP 61–3 (Xangari)

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

Abstract

We study large time asymptotics of solutions to the Korteweg–de Vries–Burgers equation
$$ u_{t} + uu_{x} - u_{{xx}} + u_{{xxx}} = 0,x \in {\text{R}},t > 0. $$
We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u 0H s (R) ∩ L 1 (R) , where \( s > - \frac{1} {2}, \) then there exists a unique solution u (t, x) ∈ C ((0,∞) ;H (R)) to the Cauchy problem for the Korteweg–de Vries–Burgers equation, which has asymptotics
$$ u{\left( t \right)} = t^{{ - \frac{1} {2}}} f_{M} {\left( {{\left( \cdot \right)}t^{{ - \frac{1} {2}}} } \right)} + o{\left( {t^{{ - \frac{1} {2}}} } \right)} $$
as t → ∞, where f M is the self–similar solution for the Burgers equation. Moreover if xu 0 (x) ∈ L 1 (R) , then the asymptotics are true
$$ u{\left( t \right)} = t^{{ - \frac{1} {2}}} f_{M} {\left( {{\left( \cdot \right)}t^{{ - \frac{1} {2}}} } \right)} + O{\left( {t^{{ - \frac{1} {2} - \gamma }} } \right)}, $$
where \( \gamma \in {\left( {0,\frac{1} {2}} \right)}. \)

Keywords

Korteweg–de Vries–Burgers equation asymptotics for large time large initial data

MR (2000) Subject Classification

35B40 35Q53