TY - JOUR
T1 - Asymptotics for the Korteweg-de Vries-Burgers equation
AU - Hayashi, Nakao
AU - Naumkin, Pavel I.
N1 - Funding Information:
Received March 4, 2004, Accepted April 11, 2005 The work of N. H. is partially supported by Grant-In-Aid for Scientific Research (A)(2) (No. 15204009), JSPS and The work of P. I. N. is partially supported by CONACYT
PY - 2006/9
Y1 - 2006/9
N2 - We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut + uux ? uxx + u xxx = 0, x ε 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 0 ε Hs (R) ∩ L 1 (R), where s > - 1/2, then there exists a unique solution u (t, x) ε C∞ ((0,∞) ;H ∞ (R)) to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u(t) = t-1/2 fM (( ·)t-1/2) + o(t -1/2) 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(t) = t-1/2 f M (( ·)t-1/2) + O(t-1/2-γ) where γ ε (0,1/2).
AB - We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut + uux ? uxx + u xxx = 0, x ε 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 0 ε Hs (R) ∩ L 1 (R), where s > - 1/2, then there exists a unique solution u (t, x) ε C∞ ((0,∞) ;H ∞ (R)) to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u(t) = t-1/2 fM (( ·)t-1/2) + o(t -1/2) 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(t) = t-1/2 f M (( ·)t-1/2) + O(t-1/2-γ) where γ ε (0,1/2).
KW - Asymptotics for large time
KW - Korteweg-de Vries-Burgers equation
KW - Large initial data
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U2 - 10.1007/s10114-005-0677-3
DO - 10.1007/s10114-005-0677-3
M3 - Article
AN - SCOPUS:33746720047
VL - 22
SP - 1441
EP - 1456
JO - Acta Mathematica Sinica, English Series
JF - Acta Mathematica Sinica, English Series
SN - 1439-8516
IS - 5
ER -