Asymptotics of solutions for sub critical non-convective type equations

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

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

We study the Cauchy problem for non-linear dissipative evolution equations ℒ(ut + script N sign(u) + ℒu = 0, x ∈ ℝ, t>0 u(0,x) = u0(x), x ∈ ℝ where ℒ is the linear pseudodifferential operator ℒu = ℱ̄ ξ→x(L(ξ)û(ξ)) and the non-linearity is a quadratic pseudodifferential operator script N sign(u) = ℱ̄ ξ→x a(t, ξ, y)û(t,ξ - y)û(t,y)dy û ≡ ℱx→ξu is the Fourier transformation. We consider non-convective type non-linearity, that is we suppose that a(t,0,y) ≠ 0. Let the initial data u0 ∈ H q,0 ∩ H0, q, q> 1/2, are sufficiently small and have a non-zero total mass ∫ u0(x)dx ≠ 0, where H n,m = {φ ∈ L2∥〈x〉 m〈i∂xnφ(x)∥ L2 < ∞} is the weighted Sobolev space. Then we give the main term of the large time asymptotics of solutions in the sub critical case.

Original languageEnglish
Pages (from-to)275-308
Number of pages34
JournalMathematical Methods in the Applied Sciences
Volume28
Issue number3
DOIs
Publication statusPublished - 2005 Feb
Externally publishedYes

Keywords

  • Dissipative evolution equations
  • Large time asymptotics
  • Non-convective type
  • Sub-critical case

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)

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