On some nonlinear dissipative equations with sub-critical nonlinearities

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

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10 Citations (Scopus)

Abstract

We study the Cauchy problem for the nonlinear dissipative equations (1) {∂tu + α (-Δ)ρ/2 u + β|u| σ u + γ|u|x u = 0, x ∈ Rn,t > 0, u(0,x) = u0(x), x ∈ Rn, where α,β,γ ∈ C, Re a > 0, ρ > 0, x > σ > 0. We are interested in the critical case, σ = ρ/n and sub critical cases 0 < σ < ρ/n. We assume that the initial data u0 are sufficiently small hi a suiatble norm, |∫u0 (x) dx| > 0 and Reβδ(α,p,σ) > 0, where δ(αρ, σ) = ∫|G(x)|σ(x)dx and G (x) = ℱ-1e- α|ξ|ρ. In the sub critical case we assume that CT is close to ρ/n. Then we prove global existence in time of solutions to the Cauchy problem (1) satisfying the time decay estimate δ(α,ρ, σ) ∫|G(x)σ G(x)dx ||u(t)||L ∞ ≤{(C(1 +t)-1/σ(log(2+t)-1/σif σ = ρ/n, C (1+t)-1/σif σ ∈(0, ρ/n).

Original languageEnglish
Pages (from-to)135-154
Number of pages20
JournalTaiwanese Journal of Mathematics
Volume8
Issue number1
DOIs
Publication statusPublished - 2004
Externally publishedYes

Keywords

  • Nonlinear dissipative equations
  • Sub-critical nonlinearities

ASJC Scopus subject areas

  • Mathematics(all)

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