Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

Abstract

We study asymptotics around the final states of solutions to the nonlinear Klein-Gordon equations with quadratic nonlinearities in two space dimensions (∂t2 - Δ + m2) u = λ u2, (t, x) ∈ R × R2, where λ ∈ C. We prove that if the final states u1+ ∈ Hfrac(q, q - 1)4 - frac(4, q) (R2) ∩ Hfrac(5, 2), 1 (R2) ∩ H12 (R2),u2+ ∈ Hfrac(q, q - 1)3 - frac(4, q) (R2) ∩ Hfrac(3, 2), 1 (R2) ∩ H11 (R2), and {norm of matrix} u1+ {norm of matrix}H12 + {norm of matrix} u2+ {norm of matrix}H11 is sufficiently small, where 4 < q ≤ ∞, then there exists a unique global solution u ∈ C ([T, ∞) ; L2 (R2)) to the nonlinear Klein-Gordon equations such that u (t) tends as t → ∞ in the L2 sense to the solution u0 (t) = u1+ cos (〈 i ∇ 〉m t) + (〈 i ∇ 〉m- 1 u2+) sin (〈 i ∇ 〉m t) of the free Klein-Gordon equation.

Original languageEnglish
Pages (from-to)3826-3833
Number of pages8
JournalNonlinear Analysis, Theory, Methods and Applications
Volume71
Issue number9
DOIs
Publication statusPublished - 2009 Nov 1

Keywords

  • Nonlinear Klein-Gordon equations
  • Quadratic nonlinearity
  • Two space dimensions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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