Large Time Asymptotics of Solutions to the Generalized Korteweg-de Vries Equation

Nakao Hayashi, Pavel I. Naumkin

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32 Citations (Scopus)


We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Korteweg-de Vries (gKdV) equationut+(uρ-1u)x+13uxxx=0, wherex,t∈Rwhen the initial data are small enough. If the powerρof the nonlinearity is greater than 3 then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. More precisely, we show that the solutionu(t) satisfies the decay estimate u(t)≤C(1+t) -(1/3)(1-1/β)forβ∈(4,∞], uux(t)L∞≤Ct-2/3(1+t)-1/3and using these estimates we prove the existence of the scattering stateu+∈L2such that u(t)-U(t)u+L2≤Ct-(ρ-3)/3for any small initial data belonging to the weighted Sobolev spaceH1,1={f∈L2; (1+x2)1/2(1-∂2x) 1/2fL2<∞}, whereU(t) is the Airy free evolution group.

Original languageEnglish
Pages (from-to)110-136
Number of pages27
JournalJournal of Functional Analysis
Issue number1
Publication statusPublished - 1998 Oct 20


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