Asymptotics in the critical case for Whitham type equations

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin

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Consider the Cauchy problem for nonlinear dissipative evolution equations {(ut + N (u, u) + L u = 0,, x ∈ R, t > 0,; u (0, x) = u0 (x),, x ∈ R,) where L is the linear pseudodifferential operator L u = over(F, -)ξ → x (L (ξ) over(u, ̂) (ξ)) and the nonlinearity is a quadratic pseudodifferential operator N (u, u) = over(F, -)ξ → xR A (t, ξ, y) over(u, ̂) (t, ξ - y) over(u, ̂) (t, y) d y,over(u, ̂) ≡ Fx → ξ u is direct Fourier transformation. Let the initial data u0 ∈ Hβ, 0 ∩ H0, β, β > frac(1, 2), are sufficiently small and have a non-zero total mass M = ∫ u0 (x) d x ≠ 0, here Hn, m = {φ{symbol} ∈ L2 {norm of matrix} 〈 x 〉m 〈 i ∂xn φ{symbol} (x) {norm of matrix}L2 < ∞} is the weighted Sobolev space. Then we prove that the main term of the large time asymptotics of solutions in the critical case is given by the self-similar solution defined uniquely by the total mass M of the initial data.

Original languageEnglish
Pages (from-to)2914-2933
Number of pages20
JournalNonlinear Analysis, Theory, Methods and Applications
Issue number10
Publication statusPublished - 2007 Nov 15
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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