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

  • Nakao Hayashi
  • Pavel I. NaumkinEmail author
  • Satoshi Tonegawa
Article

Abstract

We continue to study the existence of the wave operators for the nonlinear Klein–Gordon equation with quadratic nonlinearity in two space dimensions \({\left(\partial_{t}^{2}-\Delta+m^{2}\right) u=\lambda u^{2},\left( t,x\right) \in\mathbf{R}\times\mathbf{R}^{2}}\). We prove that if
$$u_{1}^{+}\in\mathbf{H}^{\frac{3}{2}+3\gamma,1}\left( \mathbf{R}^{2}\right),\text{ }u_{2}^{+}\in\mathbf{H}^{\frac{1}{2}+3\gamma,1}\left( \mathbf{R} ^{2}\right),$$
where \({\gamma\in\left( 0,\frac{1}{4}\right)}\) and the norm \({\left\Vert u_{1}^{+}\right\Vert_{\mathbf{H}_{1}^{\frac{3}{2}+\gamma}}+\left\Vert u_{2}^{+}\right\Vert_{\mathbf{H}_{1}^{\frac{1}{2}+\gamma}}\leq\rho,}\) then there exist ρ > 0 and T > 1 such that the nonlinear Klein–Gordon equation has a unique global solution \({u\in\mathbf{C}\left( \left[ T,\infty\right) ;\mathbf{H}^{\frac{1}{2}}\left( \mathbf{R}^{2}\right) \right) }\) satisfying the asymptotics
$$\left\Vert u\left( t\right) -u_{0} \left( t\right) \right\Vert _{\mathbf{H}^{\frac{1}{2}}} \leq Ct^{-\frac{1}{2}-\gamma}$$
for all t > T, where u 0 denotes the solution of the free Klein–Gordon equation.

Mathematics Subject Classification (2000)

35Q55 35B40 35P25 81Q05 

Keywords

Nonlinear Klein–Gordon equations Quadratic nonlinearity Two space dimensions 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Nakao Hayashi
    • 1
  • Pavel I. Naumkin
    • 2
    Email author
  • Satoshi Tonegawa
    • 3
  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityOsaka, ToyonakaJapan
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMoreliaMexico
  3. 3.Department of Mathematics, College of Science and TechnologyNihon UniversityTokyoJapan

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